Mechatronics by Analogy and Application to Legged Locomotion

Transcription

1 Mechatronics by Analogy an Application to Legge Locomotion by Victor Ragusila A thesis submitte in conformity with the requirements for the egree of Doctor of Philosophy Grauate Department of Aerospace Science an Engineering University of Toronto Copyright by Victor Ragusila 2016

2 Mechatronics by Analogy an Application to Legge Locomotion Victor Ragusila Doctor of Philosophy Grauate Department of Aerospace Science an Engineering University of Toronto 2016 Abstract A new esign methoology for mechatronic systems, ubbe as Mechatronics by Analogy (MbA), is introuce an applie to esigning a leg mechanism. The new methoology argues that by establishing a similarity relation between a complex system an a number of simpler moels it is possible to esign the former using the analysis an synthesis means evelope for the latter. The methoology provies a framework for concurrent engineering of complex systems while maintaining the transparency of the system behaviour through making formal analogies between the system an those with more tractable ynamics. The application of the MbA methoology to the esign of a monopo robot leg, calle the Linkage Leg, is also stuie. A series of simulations show that the ynamic behaviour of the Linkage Leg is similar to that of a combination of a ouble penulum an a spring-loae inverte penulum, base on which the system kinematic, ynamic, an control parameters can be esigne concurrently. ii

3 The first stage of Mechatronics by Analogy is a metho of extracting significant features of system ynamics through simpler moels. The goal is to etermine a set of simpler mechanisms with similar ynamic behaviour to that of the original system in various phases of its motion. A moular bon-graph representation of the system is etermine, an subsequently simplifie using two simplification algorithms. The first algorithm etermines the relevant ynamic elements of the system for each phase of motion, an the secon algorithm fins the simple mechanism escribe by the remaining ynamic elements. In aition to greatly simplifying the controller for the system, using simpler mechanisms with similar behaviour provies a greater insight into the ynamics of the system. This is seen in the secon stage of the new methoology, which concurrently optimizes the simpler mechanisms together with a control system base on their ynamics. Once the optimal configuration of the simpler system is etermine, the original mechanism is optimize such that its ynamic behaviour is analogous. It is shown that, if this analogy is achieve, the control system esigne base on the simpler mechanisms can be irectly implemente to the more complex system, an their ynamic behaviours are close enough for the system performance to be effectively the same. Finally it is shown that, for the employe objective of fast legge locomotion, the propose methoology achieves a better esign than Reuction-by-Feeback, a competing methoology that uses control layers to simplify the ynamics of the system. iii

4 Acknowlegements First an foremost I woul like to thank my supervisor, Prof. M.R. Emami. His support in clarifying ieas, unerstaning how to approach them in a scientific an organize way an how to communicate them effectively was priceless. He was always supportive, pushing me an always proviing very valuable feeback. I also want to thank my Doctoral Examination Committee members, Prof. Gabriele D Eleuterio an Prof. Tim Barfoot. Their avice was treasure in making this thesis a reality. The Space Mechatronics group provie both emotional support an clear, concise an tough feeback, especially when rehearsing my unorganize presentations. Thank you for all your frienship an help, in the past seven years an in many more to come. My parents stoo by my sie all the way here, an this woul have not been possible without them, for which I will be always grateful. Lastly, to Mimi, for putting up with me, an being amazing. iv

5 Contents Abstract... ii Acknowlegements... iv List of Tables... vii List of Figures... viii 1 Introuction Motivation Alternative mechatronics methoologies Choosing legge robots as a test case for mechatronics methoologies Contributions Alternative approaches to the problem of legge robots Static Stability Zero Moment Point (ZMP) Limit Cycle Walking (LCW) Capture Point Spring Loae Inverte Penulum (SLIP) Swing leg control Linkage Leg case stuy Linkage Leg mechanism Simulating the Linkage Leg Verification of Linkage Leg simulation Mechatronics by Analogy an application to legge robotics Mechatronics by Analogy Bon Graph Moeling Analogous System Concurrent Optimization Dimensional Analysis Bon graph moeling stage applie to Linkage Leg Bon graph moel of the leg mechanism Bon graph moel simplification Comparison between Linkage Leg an the simple mechanisms Bon graph stage conclusions Analogous System concurrent optimization v

6 3.3.1 Robust trajectory control Dimensional analysis stage MbA Linkage Leg conclusions Comparison between Mechatronics-by-Analogy an Reuction-by-Feeback General escription of Reuction-by-Feeback methoology Reuction-by-Feeback applie to the Linkage Leg Comparison between MbA Linkage Leg an RbF Linkage Leg Disturbance rejection comparison between MbA an RbF Linkage Legs Conclusion an further irections References Appenix: Dimensional Analysis vi

7 List of Tables Table 3.1 Variables for the test cases stuie for the bon graph simplification stage Table 3.2 Geometric parameters for the linkage leg use in bon graph simulation Table 3.3 Initial conitions for stance an swing phase bon graph simulations Table 3.4 Results of the ynamic element simplification (DES) step Table 3.5 Analogous System Optimization variables Table 3.6 Analogous System Optimization constraints Table 4.1 Reuction-by-Feeback optimization variables Table A.1 Variables for imensional analysis optimization vii

8 List of Figures Figure 1.1 Foot Place Estimator visualization... 7 Figure 1.2 SLIP moel illustration... 8 Figure 1.3 Passive knee retraction using hip torque... 9 Figure 1.4 Four bar linkage leg using passive knee retraction... 9 Figure 2.1 Linkage Leg mechanism Figure 2.2 Linkage Leg gait Figure 2.3 Linkage Leg hybri system, compose of two continuous motion phases an two transition events Figure 2.4 Graphical representation of the coupling between the hip an toe velocities of the Linkage Leg Figure 2.6 Position an angle of the robot boy comparison between the Linkage Leg moel an Simulink moel. The iscontinuous moel in Figure 2.3 is able to approximate the behaviour of the Linkage Leg Figure 2.5. Hip an knee angle comparison between the Linkage Leg moel an the Simulink moel. The transition functions in Eq. 2.5 are able to closely moel the touchown an takeoff events Figure 3.1 Mechatronics by Analogy methoology Figure 3.2 Diagram of the Bon Graph Moelling phase Figure 3.3 Bon graph representation of a rigi boy ynamics Figure 3.4 Bon graph representation of Linkage Leg, consisting of five soli boy moels from Fig Figure 3.5 Trajectories of the knee an hip joint of the bon graph moel Figure 3.6 Swing phase joint angle results for DES step, Case 3. It can be seen that the last two simplification levels, 5 an 6, iffer greatly from the original behaviour. As such, simplification level 4 is the one use in the IBS step Figure 3.7 Stance phase joint angle results for DES step, Case 3. It can be seen that all simplification levels have similar behaviour, so all the ynamic elements of the leg can be eliminate an the behaviour will be close to that of the original leg Figure 3.8 Swing phase, simplification level four Figure 3.9 Swing phase, simplification level five viii

9 Figure 3.10 Stance phase, simplification level six Figure 3.11 Comparison between Linkage Leg an ouble penulum Figure 3.12 Comparison between Linkage Leg an SLIP moel Figure 3.13 Diagram of the Analogous System Concurrent Optimization phase Figure 3.14 Analogous System Figure 3.15 Three steps of the optimal AS system Figure 3.16 Comparison of the behaviours of the initial AS system (otte line) an the optimize AS system (soli line) for one step Figure 3.17 AS two-layer control strategy Figure 3.18 Diagram of Dimensional Analysis stage Figure 3.19 Parallel Linkage Leg (left) an SLIP-like Linkage Leg (right) Figure 3.20 Gait of the optimally analogous Linkage Leg Figure 3.21 Comparison between the behaviours of the optimal AS system an the ES (LL) optimally analogous to it Figure 4.1 Reuction-by-Feeback Linkage Leg Figure 4.2 Reuction-by-Feeback Linkage Leg gait Figure 4.3 Reuction-by-Feeback Linkage Leg hip trajectory an joint torques Figure 4.4 Reuction-by-Feeback Linkage Leg virtual joint constraints Figure 4.5 MbA Linkage Leg compare with RbF Linkage Leg Figure 4.6 MbA Linkage Leg compare with RbF Linkage Leg for a 0.1m step isturbance ix

10 1 Introuction 1.1 Motivation The notion of mechatronics has evolve to a systematic esign paraigm for creating synergy an proviing catalytic impacts on iscovering simpler solutions to traitionally complex problems. The physical artifacts of such a esign philosophy, often referre to as mechatronic systems, emonstrate a seamless integration of mechanical, electrical an software constituents, in a sense that their characteristics are all specifie concurrently uring the esign phase. Concurrent engineering of such multiisciplinary systems, however, is no trivial task, for it most likely results in a large number of objective an constraint functions that must be taken into account simultaneously with a great number of esign variables. Shoul one follow a formal optimization approach, the multi-objective constraine optimization problem with large number of functions an variables can be quite challenging. The motivation behin this work is to fin an alternative solution to the problem of complex, multi-isciplinary systems, which results in a more intuitive an transparent system behaviour. The main goals are to fin unerstan the system ynamics in a simple an intuitive way, which is useful for control esign, an to be able to construct a system with esirable behaviour for a variety of situations. This thesis introuces a new concurrent esign methoology, calle Mechatronics by Analogy, that aresses the problem of system complexity in esign by i) fining simpler mechanisms analogous to the original complex system, ii) optimizing such simpler mechanisms with the controller concurrently, an iii) esigning the parameters of the original system such that it behaves similarly to the optimize simpler mechanisms with the same controller. The methoology offers qualitative an quantitative avantages over alternative methos. The qualitative avantage is that the simpler systems use for control esign are real-life mechanisms, which capture non-linear effects while can be intuitively unerstoo an stuie, an aitionally enjoy effective controllers alreay evelope for them. The quantitative avantage is that the simplification of the system behaviour occurs at the esign level, not at the control level. As such, a higher egree of synergy can be achieve between the ifferent subsystems. 1.2 Alternative mechatronics methoologies The eman for higher performance at lower cost has le to methoologies that consier concurrently the mechanical, electrical an control systems, an attempt to fin synergies between 1

11 these subsystems, which has been emonstrate to generate better an previously unattainable performance [1]. The challenge is to consier the large number of variables an various objectives which belong to a number of ifferent isciplines. A number of approaches exist to solve this problem, as will be iscusse below. One possibility is to evelop better multiisciplinary optimization (MDO) algorithms that are able to eal with a large esign space. Some involve evolutionary algorithms use for parallel robot arms [2] an genetic algorithms use for esigning reconfigurable robots [3] [4], parallel manipulators [5] an mechatronic systems [6]. The esign space can be simplifie by employing a stage optimization proceure [7] or by lowering the imensional space of the system [8] using approximations. A number of concurrent engineering approaches have been suggeste for mechatronic systems, which aress the challenges of multi-objective constraine optimization problems while also consiering subjective aspects of the esign. For instance, a concurrent system evaluation moel is propose in [9], base on three inices (efine as intelligence, flexibility, an complexity [9]), an it is formulate using t-norm an mean operators. Another evaluation moel is suggeste in [10] base on Mechatronics Design Quotient (MDQ), where a nonlinear fuzzy integral is use for the aggregation of esign criteria. An alternative concurrent esign methoology is presente in [11], introucing the notion of Holistic Concurrent Design (HCD), where the subjective aspects of multiisciplinary systems esign are capture through fuzzy operators, along with the quantitative system performances using a holistic system moeling base on bon graphs. Another notable effort is the Design for Control (DFC) methoology [12], which prescribes that control parameters be esigne concurrently with the structure parameters, so that by simplifying the unerlying system ynamics simple controllers can be employe successfully. In aition, a number of a hoc techniques for mechatronics have been reporte in the literature, which are innovative esign schemes for specific applications such as robotics (e.g., see [13]). Metho of Imprecision (MoI) [14] is a notable preliminary esign methoology is able to consier imprecision in esign an is use to efine a set of esign preferences regaring the esign space optimization traeoffs [15]. Methos exist to simplify the close loop ynamics of the robots using a control layer. The metho presente in [16] requires a fully actuate leg achieve forces upon the boy similar to those generate by any esirable mechanism. The forces in the joints of the robot are obtaine using the 2

12 reverse Jacobian an the forces generate by the virtual moel. This also requires actuators able to achieve force or torque control of the joints, which is a ifficult to achieve from a mechanical construction point of view [17]. Another metho is presente that achieves esirable close loop behaviour using a control layer [18]. There are examples shown for achieving similarity to simpler mechanism for a simple test case, but not for the full robot. This approach also requires fully actuate robotic legs, but it oes not require the complex force feeback capable joints. The main issue with these methos is that they require fully actuate robot legs an power to achieve the require close-loop ynamics because they attempt to cancel the natural ynamics of the leg instea of utilizing them. Moreover, the range of mechanisms they are able to emulate is restricte [18], an require extra actuator power an, in the case of [16], actuators able to achieve force feeback control. 1.3 Choosing legge robots as a test case for mechatronics methoologies The Mechatronics by Analogy methoology is riven by the nees in the legge robots research community. Legge robots offer an interesting challenge for mechatronic methoologies. In orer to simplify the control of robotic legs an make the behaviour more tractable an intuitive, the physical moel of a leg mechanism must be simple. Simple moels, such as the inverte an ouble penulums are use to approximate the legs of walking robots [19], an the Spring Linear Inverte Penulum (SLIP) moel is use to approximate the behaviour of running an hopping robots [20]. The behaviour of such moels, as well as the controllers evelope for them, have been extensively stuie both in the fiel of robotics an biological walkers an hoppers [21]. However, the control esign can become far too complex, thus challenging, if the mechanical esign of the legs goes beyon such simple moels [22], an consequently it will be ifficult to take avantage of the wealth of information alreay available for the simpler moels. 1.4 Contributions The thesis has a number of important contributions. First, a new mechatronics methoology, ubbe as Mechatronics by Analogy (MbA), is propose an evelope. This methoology is first introuce in [23]. The MbA methoology offers a number of avantages compare with the existing methoologies: Quantitative avantages: 3

13 o The methoology oes not require aitional control layers, making the control simpler an faster than alternative, control intensive methos, as the one in Chapter 4. o No nee for complex force-feeback capable actuators. o Better performance can be achieve by exploiting synergies at the esign phase. Qualitative avantages o Control is base on well-unerstoo, real-life mechanisms. o The simplifie systems capture the non-linear effects of the original system. o Simplification occurs at the esign level, not at the control level. This results in a more intuitive unerstaning of the system. The MbA methoology is etaile through the esign of a legge robot. A number of important contributions are mae: A simplification proceure base on bon graphs is evelope, an it is shown to successfully simplify a robot leg for ifferent motion phases [24]. The simpler system is optimize using a concurrent optimization approach that avois complex issues, such as impulses ue to contact between the robot an the groun [25]. A leg is evelope using a imensional analysis metho, which achieves similar ynamics to those of a esire set of simpler systems [26]. The final contribution of the presente work is a comparison between the MbA methoology an a competing proven methoology. This comparison is mae by esigning a legge robot using both methoologies, with the aim of achieving fast legge locomotion. The MbA esigne robot is able to achieve a higher top spee using less torque, while the competing robot is able to eal with isturbances slightly better. Publications: - V. Ragusila an M. R. Emami, "A mechatronics approach to legge locomotion," in Avance Intelligent Mechatronics, Montreal, V. Ragusila an M. R. Emami, "A novel robotic leg esign with hybri ynamics," Avance Robotics, vol. 27, no. 12, pp , V. Ragusila an M. R. Emami, "Moelling of a robotic leg using bon graphs," Simulation Moelling Practice an Theory, vol. 40, pp ,

14 - V. Ragusila an M. R. Emami, "Mechatronics by Analogy an Application to Legge Locomotion," Mechatronics, accepte, July V. Ragusila an M. R. Emami, " Reuction by Design vs. Reuction by Feeback: A Benchmark Stuy for Legge Robotics," Mechatronics, submitte September Alternative approaches to the problem of legge robots To unerstan where the current methoology fits comparatively with existing methos that aress the problem of walking robots a review of existing approaches is presente below. Although the fiel of legge robots is vast an a lot of robots are being use in the inustry for traveling insie pipes, winow washing or heavy transport, this review will eal mainly with monopo an bipe robots use in acaemia to stuy the basic features of legge locomotion. For a review of other legge robots, [27] is a very goo reference. The fiel of legge robots is new, with the first experiments being conucte at Wasea University in the 60s, there are many approaches efining how legge locomotion can be achieve. These paraigms represent the basic iea behin the esign of the robot [28] an some have been efine an given a name after robots have been built base on them. There are two major types of bipeal or monopo robots, planar an 3D. Planar robots are restricte to movement in the sagittal plane an cannot move laterally, roll or yaw [20]. They are use mainly to test new theories in low cost an simple robots, while 3D robots o not have such restrictions Static Stability First use in 60s, it was the first approach which allowe bipeal robots to walk. The static stability approach states that the Center of Mass (COM) must be over the support polygon at all times, an must move slowly enough such that ynamic effects are not important [28]. Most of the control is open loop trajectory tracking, using PD controllers for each iniviual joint. The trajectory is programme in such that it keeps the COM over the support polygon. This approach is rarely use in active research, since Zero Moment Point (ZMP) has largely replace it, but it is still use in hobby bipes an in soccer playing bipes [29], which are very cheap an simple to buil, allowing amateur builers to esign their own bipes Zero Moment Point (ZMP) The Zero-Moment-Point (ZMP) locomotion paraigm is one of the most popular, being use in such robots as the ASIMO [30], QRIO [31], HRP series of humanois [32] an many others 5

15 [27] [33]. The ZMP is efine as the center of pressure of the foot. If this point is not on the ege of the support polygon, efine by the legs in touch with the groun, the robot can apply a moment on the groun an its trajectory can be controlle using classical robotic manipulator approaches [33]. This means the robot must have a foot flat on the groun at all times, increasing the complexity of trajectory calculation an severely limiting its strie length, especially preventing running gaits with aerial phases especially preventing running gaits with aerial phases without significant changes to the control algorithm [34] Limit Cycle Walking (LCW) Limit Cycle Walking was first introuce in [28] as a sequence of steps that is locally unstable but stable over a number of steps. However robots base on the LCW approach has been evelope in some form since the first hopping robots evelope by Raibert [35] an the passive ynamic walkers mae by McGeer [36]. As the efinition is quite wie, there are quite a number of robots esigne an built which fall in this efinition. The first robots that achieve LCW are the passive ynamic walkers, which consist of a series of penulums an can walk own a slope. There is no control an no actuation for these robots, but they have been proven to be stable, an to be able to eal with small isturbances [28] [36]. State of the art passive ynamic robots have an upper boy an knees an are no longer planar [36] [37]. Recently, actuation has been ae to passive ynamic robots keeping the control system if very simple. The Cornell Walker creates a small impulse to the ankle of the stance leg when the swing leg hits the groun [38] an Denise has a small moment applie to the hip joint when the next leg hits the groun [19] [39]. These robots are tune to walk at a preefine spee an they can eal with small isturbances, without the nee of actively controlle joints or any sensors besie foot switches. These robots are very energy efficient, achieving a cost of transportation similar to humans [19]. RunBot [40] has 4 servo actuators in the hip an knee joints an a fifth one in its torso, an is the fastest walking bipe for its size. Its control is base on a neural network, which ajusts the signals to the 5 actuators base on ata from foot switches an joint position sensors. The robot achieves stable cyclic walking at a esire spee an can learn to walk up a slope. A metho is propose by Hyon that achieves stable hopping motion for a monopo base on the property of Hamiltonian systems of achieving a stable orbit if energy is not change in the 6

16 system [41]. He proposes a monopo with a Spring Loae Inverte Penulum (SLIP) leg with impeance controllable hip an leg joints an which has three levels of control which can aapt to any starting conitions to achieve passive-ynamic running Capture Point The Capture Point approach allows a robot to regain stability after being subjecte to a isturbance. This concept can be evelope into a walking gait by consiering each step a isturbance, an not recovering completely from each previous step. A Capture Point (CP) is the foot placement position that allows the robot to come at a stop without any more control being applie. Capture Point is use in [42], whereas a similar concept calle Foot Place Estimator (FPE) is introuce in [43], an it is shown in Figure 1.1. Figure 1.1 Foot Place Estimator visualization [43] The stability margin for the CP approach is the number of steps it takes to reach a Capture Point. A three step margin means that the capture point is 3 steps away, an the robot cannot stop in less than three steps [42]. If the legs of the robot are not fast enough, a bipe might fall own by tripping its swing foot Spring Loae Inverte Penulum (SLIP) The SLIP moel is a very popular moel for escribing ynamic locomotion, which might exhibit running gaits with aerial phases [20]. It consists of a point mass (the boy of the robot) connecte to the groun using a linear spring which is aligne with the leg axis. The SLIP moel 7

17 has been use extensively in biology to simulate a running gait [44], an some argue that a goo efinition for running is of a motion which uses the SLIP moel to store an release energy [45]. Raibert has use the SLIP first in robotics an a series of highly successful planar an 3D monopos, bipes an quarupes have been create at the MIT Leg Lab using this moel [20] [46]. The classic SLIP moel is telescopic [47], but segmente legs with a knee or a knee an an ankle seem to give better stability an allow running at lower spees [48] [49]. Construction is easier for a segmente leg, since no heavy prismatic joints nee to be use [22]. Other legs are constructe using flexible bow-like structures [50] [51] or are approximate using linkages an springs [52]. Figure 1.2 SLIP moel illustration [47] Most SLIP control strategies are focuse on the touchown moment, when the leg touches the groun. The angle an angular velocity at which the leg hits the groun have been shown to be critical for the leg gait [41] [53]. Other methos have been able to use tune springs in the hip joint an knee joint to obtain fully or partially passive running gaits. [54] [55]. A goo overview of variations of the SLIP moel an control strategies can be foun in [56] Swing leg control The SLIP is a very popular moel for the stance phase of a legge robot, an a lot of the current research is aime at controlling an improving its behaviour. However, the ynamics of the leg uring the swing phase are also very important, both for single legge an multi-legge robots. Most robotic legs are moelle as simple penulums with rotation axis at the hip motion [39]. The perio of these simple penulum swing legs etermines the perio of stable gait for most 8

18 Figure 1.3 Passive knee retraction using hip torque [54] Limit Cycle Walking robots [36]. Aing a spring aroun the hip joint can increase the range of swing spees for these legs an also increase the robustness of the robot gait [57]. The behaviour of an actuate hip with passive knee is escribe in [58]. The hip escribes a preetermine trajectory motion to bring the swing leg ahea of the stance leg in the require time, an the knee is left passive. Once the knee straightens, usually just before groun contact, it is mechanically locke into place to prevent bouncing or vibrations. This leas to both energy efficiency an better robustness against isturbances. A four bar linkage leg esign with passive retraction, is escribe in [22], but it requires the joints to have zero impeance, which was impossible with the mechanical esign of that robot. Horse legs are theorize to have a passive catapult using elastic elements in the leg that store energy an use it to rapily contract the swing leg [59]. The catapult system allows the horse to use 100 times more power in contracting an swinging the front legs than is available in the actual muscle, an thus lower the inertia of the swing leg. Figure 1.4 Four bar linkage leg using passive knee retraction [22] 9

19 2 Linkage Leg case stuy 2.1 Linkage Leg mechanism The Linkage Leg is a novel robotic leg esign first propose by the author in [26]. The goal of the esign is to provie a structurally efficient, simple to control leg with tractable behaviour for both the swing an stance phases. To that en, the MbA methoology is use in the esign an construction of the propose leg. The propose leg esign, is compose of four links. The first link is the thigh, {O 0 O 1 O 4 } in Figure 2.1, which is connecte to the boy of the robot using the hip joint O 0. The tibia {O 1 O 2 }, foot {O 2 O 3 O 5 } an tenon {O 4 O 5 }, together with the thigh, form a four-bar linkage. The lengths a 1 a 6 together with the angles φ 1 an φ 3 etermine the geometry of the leg. Figure 2.1 Linkage Leg mechanism The Linkage Leg has two egrees of freeom. For the swing phase, they are the angle between 10

20 coorinate frames {O 0 } an {O 1 }, efine as the hip angle θ 1 an the angle between the coorinate frames {O 1 } an {O 2 }, efine as the knee angle θ 2, as shown in Figure 2.1. For the stance phase, they are the angle between coorinate frames {O 3 } an {O 2 } (θ 6 ), an the angle between the coorinate frames {O 2 } an {O 1 } (θ 7 ), as shown in Figure 3. Note that the Denavit-Hartenberg [60] notation is use for assigning the link coorinate frames of the linkage leg in both stance an swing phases. As shown in Figure 3.1, a chain (1), starting from point A an ening at point O 5, is wrappe aroun cogs at the joints O 1 an O 4. Point A is chosen such that the length of the chain remains constant as the knee angle θ 2 changes. A motor is attache to the cog at joint O 4 an controls the length of the leg uring the swing phase by actuating the chain. During the stance phase the cog at joint O 4 locks the chain preventing it from rotating aroun the cogs an forcing the elastic element (2) to exten an store energy. The leg has two actuators. The hip actuator at joint O 0 controls the angle of leg relative to the boy of the robot. The knee actuator at joint O 1 controls the length of the leg uring swing, an it is also use in parallel to the elastic element (2) to control the height of each hop. The Linkage Leg mechanism has a number of avantages over similar robotic legs. The leg employs only revolute joints, making it simpler to buil an potentially lighter than legs with prismatic joints [50]. The tibia an tenon segments are loae only in compression an tension, respectively, making them easier to esign an lighter to buil. Another significant avantage is that the elastic element (2) nees to act at tension only, allowing the use of rubber or latex that have a higher energy storing capacity per unit mass compare to steel springs. Further, the propose Linkage Leg can change its knee angle at touchown easily by changing the timing for locking the string with no nee for mechanical ajustments. The most significant avantage of the Linkage Leg is that its ynamics can be tune to a SLIP moel for the stance phase an to a specific ouble penulum for the swing phase, using the approach that will be iscusse in the sequel. This allows for a simple an intuitive metho of unerstaning an efining the ynamics of the leg, as well as the use of existing control strategies for the SLIP an ouble penulum [61] [20]. The complexity of achieving a esirable hopping gait is, therefore, shifte from the online control to the offline mechanical esign of the leg. Although the mechanical esign of the leg 11

21 Takeoff Touchown Takeoff Figure 2.2 Linkage Leg gait requires more effort in assigning the physical parameters so that a esire ynamics is achieve, the online control will become simpler. This approach, which can be characterize as reuctionby-esign [62], is in contrast with reuction-by-feeback approaches [61] [18], where the complexity lies in the online control. Unlike reuction-by-feeback approaches where the goal is to esign a (complex) controller that makes the overall close-loop system achieve a simpler ynamics, the propose approach aims at esigning a leg mechanism that achieves a simpler ynamics by optimizing its physical parameters. 2.2 Simulating the Linkage Leg The Linkage Leg robot is esigne to achieve a repeatable hopping gait, as shown in Figure 2.2. This gait starts at takeoff, when the leg breaks contact with the groun. The swing phase starts at takeoff an lasts until touchown, when the leg contacts the groun. The stance phase lasts while the leg is in contact with the groun, after which the gait repeats itself. The ynamic equations of the Linkage Leg are show in Eq. (2.1)-(2.4): x sw = f sw (x sw, τ 1,sw, τ 2,sw ) (2.1) x st = f st (x st, τ 1,st, f L,st ) (2.2) x sw = [q sw q sw ], q sw = [x 0 y 0 θ B θ 1 θ 2 ] (2.3) x st = [q st q st ], q sw = [θ L L st θ H ] (2.4) 12

22 x st,i = sw st (x sw,f ) Swing Phase x sw = f sw (x sw, τ 1,sw, τ 2,sw ) Stance Phase x st = f st (x st, τ 1,st, τ 2,st ) x sw,i = st sw (x sw,i ) x sw,i Figure 2.3 Linkage Leg hybri system, compose of two continuous motion phases an two transition events The gait is simulate using two functions: x sw = f sw (x sw, τ 1,sw, τ 2,sw ) to simulate the swing phase an x st = f st (x st, τ 1,st, f L ) to simulate the stance phase, as shown in Figure 2.3. The inputs to these functions are the states of the system uring swing (x sw ) an stance (x st ), together with the torques of the controlle egrees of freeom, τ 1, τ 2. The outputs of the two phase functions are the erivatives of the state space vectors. 13

23 The swing phase state vector, x st is show in Eq. 2.3, where x 0 is the horizontal istance travele by the hip joint from the previous takeoff location, y 0 an θ B are the height an angle of the hip joint with respect to the groun an θ 1 an θ 2 are the controlle egrees of freeom of the Linkage Leg mechanism. The stance phase state space vector is shown in Eq. 2.4, where θ L is the angle of the virtual leg (efine as the segment connecting the toe O 3 to the hip joint O 1 ) with respect to the groun, L st is the length of the virtual leg an θ H is the angle of the boy with respect to the virtual leg. The stance an swing functions are presente below in etail: f sw = f st = [ [ 1 M sw τ 1 ( [ τ 2 ] C sw (q sw, q sw )q sw g sw (q sw ) ) q sw ] 0 M 1 st ([ τ 2 ] C st (q st, q st )q st g st (q st )) τ 1 q sw ] (2.5) (2.6) The matrices M sw an M st are the mass matrix of Linkage Leg an robot boy mechanism, the matrix C sw incorporates the Coriolis an centrifugal effects, an g sw is the column vector expressing the effects of gravity. The values in the torque column vector are τ 1, the torque in the hip joint, an τ 2, the torque in the knee joint, which are etermine by the control system as shown in Section Two transformation functions are use at the takeoff an touchown moments to transition the system between the two motion phases, an thus represent the effects of impulses without requiring a etaile knowlege of the groun properties or the elasticity of the system. The initial state of the stance phase, x st,i, is efine as a function of the final state of the swing phase, namely x st,i = sw st (x sw,f ). Similarly, the initial state of the swing phase, x sw,i, is efine as a function of the final state of the stance phase, i.e., x sw,i = st sw (x st,f ) in Eq. (2.7). 14

24 tan 1 (y 0,sw,f (x 0,sw,f x toe,sw,f )) (x 0,sw,f x toe,sw,f ) y 0,sw,f sw st = π φ offset θ 1,sw,f ((x 0,sw,f x toe,sw,f )y 0,sw,f y 0,sw,f x 0,sw,f) ((x 0,sw,f x toe,sw,f ) 2 + y 0,sw,f ) 2 ((x 0,sw,f x toe,sw,f )x 0,sw,f + y 0,sw,f y 0,sw,f) (x 0,sw,f x toe,sw,f ) y 0,sw,f [ θ 1,sw,f ] st sw = L st,f cos θ L,st,f L st,f sin θ L,st,f θ L,st,f + θ H,st,f π φ offset θ H,st,f (L st,f L st,i )C L st,f cos θ L,st,f L st,f sin θ L,st,f θ L,st,f L st,f sin θ st,l + L st,f cos θ L,st,f θ L,st,f θ L,st,f + θ H,st,f (2.7) [ θ H,st,f L st,fc ] The parameter C in (7) represents the coupling between the rate of change of virtual leg length, L, an the knee joint, θ 2, calculate from the geometry of the Linkage Leg. The parameter C is calculate using the point G, efine as the intersection of segments O 1 O 2 an O 4 O 5 an shown in Figure 2.4. The point G becomes the instantaneous center of rotation of the soli boy {O 2 O 3 O 5 }. As such, the irection of the instantaneous velocity of the toe O 3 cause by a change in the knee angle is perpenicular to the segment O 2 G, an its magnitue is θ 2 Consequently, C = a 2 O 3 G O 2 G. a 2 O 3 G O 2 G. 15

25 Figure 2.4 Graphical representation of the coupling between the hip an toe velocities of the Linkage Leg. 2.3 Verification of Linkage Leg simulation The swing an stance phase functions, f sw an f sw together with the transition functions st sw an sw st are able to simulate the behaviour of the Linkage Leg system. However, the assumption is mae that the boy velocity oes not change at touchown, but instea the velocities of the hip an knee egrees of freeom, θ 1 an θ 2 have an instantaneous change at touchown. Of course a real system oes not experience instantaneous changes of velocities. In orer to valiate the assumptions about the takeoff an touchown transitions, a Simulink moel is use to simulate the Linkage Leg an investigate the touchown an takeoff moments. SimMechanics is a soli boy simulation package for Simulink. The Linkage Leg moel is show in Figure 2.5. It consists of five soli boies, four for the Linkage Leg segments an one for the boy of the robot, linke together by revolute joints. The toe point is connecte to a groun moel that calculates the force the groun woul apply any time the point ips below the groun level. The function use for the groun moel is escribe in [63] an uses the same parameters obtaine there. The torque profiles τ 1 an τ 2 use in the gait shown in Figure 2.2 are applie to the hip an knee joints of the Simulink moel. The results are compare below with those obtaine using the 16

26 functions escribe in section 2.2. At touchown (time 0.6s) the Linkage Leg moel transition function sw st changes the swing state space to the stance state space, an the Simulink moel experiences groun contact. As it can be seen in Figure 2.5 the Linkage Leg moel is iscontinuous, using the matrices in Eq. (2.7) to approximate the impulses at touchown an takeoff, whereas the Simulink moel is continuous an both the hip angle 1 an 2 exhibit vibrations ue to the contact with the groun, similarly to the touchown vibrations shown in [63] [40]. The iscontinuous Linkage Leg moel was able to closely approximate the more realistic continuous Simulink moel for the knee an joint trajectories, as well as for the behaviour of the robot boy, shown in Figure 2.6. As such, the transition functions in Eq. 2.5 are able to closely approximate the complex touchown an takeoff events, an the simple, iscontinuous moel from Figure 2.3 is a suitable approximation of the realistic Linkage Leg behaviour. The iscontinuous moel will be use in the rest of the thesis to simulate the behaviour of the Linkage Leg (eg) Simulink Moel 1 Linkage Leg 1 1 /t (eg/sec) Simulink Moel 1 /t Linkage Leg 1 /t 2 (eg) time (s) Simulink Moel 2 Linkage Leg time (s) Touchown instantaneous change 2 /t (eg/sec) Simulink Moel 2 /t Linkage Leg 2 /t time (s) time (s) Figure 2.5. Hip an knee angle comparison between the Linkage Leg moel an the Simulink moel. The transition functions in Eq. 2.5 are able to closely moel the touchown an takeoff events. 17

27 hip horizontal position (m) forwar position Simulink moel forwar position Linkage Leg hip horizontal velocity forwar velocity simulink moel forwar velocity Linkage Leg time (s) time (s) hip vertical position (m) vertical position Simulink moel vertical position Linkage Leg time (s) hip vertical velocity vertical velocity simulink moel vertical velocity Linkage Leg time (s) Figure 2.6 Position an angle of the robot boy comparison between the Linkage Leg moel an Simulink moel. The iscontinuous moel in Figure 2.3 is able to approximate the behaviour of the Linkage Leg. 18

28 3 Mechatronics by Analogy an application to legge robotics 3.1 Mechatronics by Analogy Mechatronics by Analogy (MbA) aresses the challenge of concurrent esign by fining simple mechanisms analogous to a more complex mechanical system, optimizing the simpler Figure 3.1 Mechatronics by Analogy methoology 19

29 mechanisms, an then esigning the control system aroun these simpler mechanisms. The systems nee not be equivalent, an a egree of analogy is introuce, similar to the imperfect analogy use in [64]. The closer the two systems are, the better the controller performance, but a perfect equivalence is neither neee nor possible in most cases. Furthermore, MbA offers techniques to improve the similarity between the simple mechanisms an the actual mechanical implementation of the system. The three stages of the MbA methoology are shown in Figure 3.1 an etaile in the following three sections. The process begins with a esire concept of the original system, here referre to as Emulate System (ES). A bon graph moel of the system is then evelope, an the moel is simplifie base on the universal notion of energy to fin a set of simpler mechanism, calle the Analogous System (AS), which relevantly emulates the ES. Next, the AS is etail-esigne concurrently with the control system to fin the optimal system parameters for the esire behaviour, obtaining an optimal AS. The control system may be taken from the literature or etermine using other methos, as explaine in sections an 3.3. Finally, the parameters of the ES are erive, through the imensional analysis stage, such that the ES system behaves similarly to the optimum AS. If this similarity is achieve, then the controller evelope for the AS system can be applie without moifications to the ES. Hence, the complex task of optimizing the physical an control parameters of the ES is replace by the simpler tasks of optimizing the parameters for the AS together with the control system, an then fining the ES parameters that achieve similarity. As it will be emonstrate in Section 4, this approach is easier to achieve an leas to better results than tackling the ES optimization irectly, for the case of a robotic linkage leg esign. MbA methoology is ieally suite for systems that have multiple phases. A phase is efine as a set of conitions or constraints that specify the system operation. The Linkage Leg robot case stuy has two motion phases, the stance phase, when the leg is in contact with the groun, an the swing phase, when the leg is not in contact with the groun. As will be shown in sections 3.2 an 3.3, the MbA methoology evelops separate simpler mechanisms an controllers for each motion phase, making the AS concurrent optimization simpler. The imensional analysis stage is then able to fin an ES mechanism that is similar to each of the simpler mechanisms of the AS, combining the optimal behaviour of multiple mechanisms into one system, as seen in section 3.4. The concept of phases can be generalize to other mechatronic systems that nee to function in ifferent 20

30 conitions or achieve ifferent goals at ifferent times. If the system has only one phase, the MbA methoology becomes a simple simplification metho, an is less beneficial. The MbA methoology is explaine in section 3.1 for a general system, an then applie for the case stuy of a legge robot in sections Bon Graph Moeling The first stage of MbA involves analyzing the ES an fining the simplest mechanism that represents the ynamic behaviour of the original ES for each of the N motion phases of the ES. This is one through the evelopment of the bon graph moel of the ES. Bon graphs represent the ynamics of a system by simulating the power exchange between its components [65]. The system variables (force, velocity, current, voltage, etc.) are unifie into two generalize variables, the flow an effort. The power flow between components can be compute by multiplying the flow an effort of each element. The ynamic behaviour of the bon graph is compute by consiering the relation between flow an effort in each element. Further, a more complex system can be represente by simpler bon graphs linke together in a moular fashion [66]. The MbA s Bon Graph Moeling stage consists of a two-step simplification. The first step, Dynamic Element Simplification (DES), eliminates the maximum number of ynamic elements while maintaining the system behaviour nearly unchange [67]. For a bon graph representing a mechanism forme by soli boies, the ynamic elements are masses, moments of inertia, springs an ampers. For each possible combination of the system ynamic elements the bon graph is simulate, an the behaviour of a set of representative test bons is compare. The result of DES is the smallest set of ynamic elements that can still prouce a close behaviour to the original system. The secon step, Iniviual Bon Simplification (IBS), is applie to the set of elements remaining after the DES step. The goal is to fin the essential bons in the bon graph escribe by the remaining ynamic elements. For this purpose, a metric relate to the power flow of each bon is use to eliminate the bons with zero or negligible power flow [66]. The two steps of the Bon Graph Moeling stage will be further illustrate through the case stuy in Chapter Analogous System Concurrent Optimization The goal of the AS Concurrent Optimization stage is to obtain the best AS mechanism an controller for each respective motion phase. Since the AS is compose of ifferent mechanisms, 21

31 each representing a vali analogy to the ES for its corresponing motion phase, it is possible to avoi ifficult control problems, such as switching between ifferent sets of constraints ue to impact or contact events. This is achieve by esigning controllers that are require to control each AS mechanism only uring their respective motion phase, an by fining appropriate transition conitions between the motion phases. Choosing the control strategy is the first step of the AS Concurrent Optimization stage. Since the AS is compose of realistic, nonlinear mechanisms (e.g., ouble penulum, SLIP, etc.,) control strategies base on such mechanisms can be reaily implemente. The selecte control strategy is epenent on the mechanisms resulting from the Bon Graph Moeling stage, an the behaviour require from the final system. In the case of legge robots, for example, this task is simplifie, because a variety of control strategies have been propose for the group of simple mechanisms that are analogous to the Linkage Leg [20] [55] [68]. The next step is to ientify the transition conitions between motion phases. At these transitions the AS system changes between one mechanism to another. The transition conitions epen on the exact nature of the AS obtaine from the Bon Graph Moeling stage, however two rules shoul be followe: The first rule is that the potential energy of the system must not change at the transition between the ifferent mechanisms forming the AS. As such, the total mass an position of the centre of mass must remain the same for each mechanism forming the AS. The justification for this rule is that it ensures the optimize ES through the Dimensional Analysis stage is not require to be analogous to an AS forme by mechanisms with ifferent mass or moment of inertia. Otherwise, it woul be nearly impossible to efine a unique set of parameters for the ES so that it behaves similarly to all the multiple mechanisms of the AS, if the mass or location of the center of mass of these mechanisms iffers greatly. The kinetic energy or angular momentum cannot be preserve at transitions, because impacts cause energy loss in the system. The secon rule is that the rates of the egrees of freeom unaffecte by the switch of constraint conitions impose by the contact are kept constant, while the rates of the other egrees of freeom experience an instantaneous change accoring to the specific impact or contact. This will be illustrate in chapter 3 by analyzing the specific cases of the touchown an takeoff transitions between the AS mechanisms. Once the transition conitions have been ientifie, each AS mechanism an control parameters can be concurrently optimize. Such a multiisciplinary concurrent optimization can 22

32 be solve by a variety of algorithms, ue to its simple structure an low number of variables [3] [4] [69], whereas the same algorithms woul likely have ifficulty tackling with the original ES, as it will be illustrate in the following sections Dimensional Analysis The purpose of the Dimensional Analysis stage is to fin those parameters of the ES which result in the closest behaviour to that of the previously optimize AS. This is achieve by employing the concept of ynamic similarity an requiring the ES an AS to be approximately ynamically similar, i.e., analogous. The motivation for this approach is that if a sufficiently close ynamic similarity is achieve between the ES an AS, the controller evelope concurrently with the AS uring the secon stage of MbA can be applie to the ES, without moifications while maintaining the same synergistic performance. The general concept of ynamic similarity is explaine in [70]. Dynamic similarity starts with an equation of the general form Eq. (3.1), which relates the i groups of basic variables that efine the phenomena: n = 0 (3.1) By replacing the units of Eq. (3.1) with appropriate values the groups i are replace by imensionless groups i, calle Pi-factors, which represent the class of phenomena inepenently of the basic variables. If the constitutive Pi-factors are equal for a class of phenomena, then these phenomena are consiere ynamically similar. To apply the concept of ynamic similarity to the esign of the ES, the Lagrangian equation of the ES is use as Eq. (3.1). The Pi-factors are obtaine by replacing the units of length, time an energy as shown in [71]. An optimization proceure is use to obtain the parameters of the ES such that its Pi-factors are equal to the corresponing Pifactors of the AS. If a perfect match is not achieve between the sets of Pi-factors, the ynamic similarity will not be exact. It is possible, however, to achieve a esirable result with an approximate similarity, i.e., analogy, as it will be shown for the case stuy of the Linkage Leg. In some cases the mechanisms forming the AS will have a ifferent set of egrees of freeom, ue to some external constraints being present in some motion phases but not in others. In such cases the Lagrangian of the ES will be transforme to the coorinates of respective AS mechanism. An example of such transformation will be shown for the stance phase of the Linkage Leg. 23

33 Figure 3.2 Diagram of the Bon Graph Moelling phase 3.2 Bon graph moeling stage applie to Linkage Leg The presente work aims at fining simple moels, using bon graphs, with approximately similar ynamic behaviour to that of a given leg mechanism for each motion. Bon graphs are omain-inepenent graphical escriptions of ynamical behaviour of physical systems. Bon graphs are base on the concept that energy exchange is a common notion to ynamic systems regarless of their physical omain [72]. Therefore, in a bon graph components are efine by their energetic behaviour; they can either supply or absorb, store or issipate, an reversibly or irreversibly transform energy [73]. Bon graphs can be use to represent complex three imensional mechanical systems [74] [75], as well as mechatronic systems [76] [77] [78] [65]. The first part of this section expresses ynamics of a leg mechanism using bon graphs in a way that it facilitates further simplifications. The secon part of the paper, Section 3.1.2, applies a combination of various simplification techniques to the bon graph moel to fin simpler representation of the system in each motion phase. The behaviour of simpler moels is also compare with that of the original moel in Section Some concluing remarks are mae in Section Bon graph moel of the leg mechanism The goal of this section is to obtain a moel for the linkage leg mechanism that can be simplifie to obtain the simple mechanisms that will form the AS. Bon graphs represent the system ynamics by consiering the power exchange between its components. The system variables such as force, velocity, current, voltage, etc., are unifie into two groups, the flow an effort. By multiplying these generalize variables, the power flow between components can be 24

34 compute. The bon graph uses the ynamic equations of its components an initial conitions to calculate the behaviour of the system [65]. Furthermore, bon graphs can be linke together in a moular fashion to represent a more complex mechanism. For example, the rigi boy in Figure 3.3 is represente using a bon graph compose of several moules. The generalize flow an effort variables correspon to velocity an force, respectively. The velocity (flow) of point O 1 is specifie using flow source components (C), an the force (effort) applie at point O 2 is represente by effort source components (D). Both velocity an force vectors are expresse in the local coorinate frame of the boy. The vectors r 1 an r 2 efine the position of points O 1 an O 2 with respect to the centre of mass of the boy in the local coorinate frame using transformer blocks (B). The rotational ynamics of the boy are represente using an inuctor component (A), an the translational ynamics of the boy are represente by two inertia components as inuctors an two effort sources in the worl coorinate frame (F). A coorinate transformation block (E) relates the velocities an forces of the boy centre of mass expresse in the worl an local coorinate frames. Using the above-mentione six blocks the planar ynamics of the rigi boy can be calculate, together with the power flow to each component. By linking a number of rigi boy moules together, the ynamics of a more complex system, such as the leg mechanism shown in Figure 2.3 can be erive. Figure 3.3 Bon graph representation of a rigi boy ynamics 25

35 The bon graph representation of the Linkage Leg is shown in Figure 3.4. The bon graph is compose of five rigi boy moules, the thigh, tibia, foot, tenon an robot boy, to which the leg is attache. Aitionally, a groun moel is use to simulate the toe (O 3 ) touching the groun. The bon graph moel of Linkage Leg was simulate in two ifferent conitions: the stance phase an swing phase, an its behaviour was compare with the moels of the same mechanism Figure 3.4 Bon graph representation of Linkage Leg, consisting of five soli boy moels from Fig

36 evelope using both MATLAB SimMechanics physical moeling as well as the m-file script, as escribe in Chapter 2. Table 3.1 Variables for the test cases stuie for the bon graph simplification stage Inertia variable Case 1 Case 2 Case 3 m 1 (kg) m 2 (kg) m 3 (kg) m 4 (kg) I 1 (kg m 2 ) I 2 (kg m 2 ) I 3 (kg m 2 ) I 4 (kg m 2 ) Table 3.2 Geometric parameters for the linkage leg use in bon graph simulation Geometric variable All cases Geometric variable All cases a 1 (m) 0.5 x 1 (m) 0.25 a 2 (m) 0.5 x 2 (m) 0.25 a 3 (m) 0.5 x 3 (m) 0.15 a 4 (m) 0.3 x 4 (m) 0.25 a 5 (m) 0.5 y 1 (m) 0.00 a 6 (m) 0.2 y 2 (m) 0.00 φ 2 (eg) 0.00 y 3 (m) 0.00 φ 3 (eg) π y 4 (m) 0.00 Table 3.3 Initial conitions for stance an swing phase bon graph simulations Vertical boy Horizontal boy θ 1 (eg) θ 1 (eg/s) θ 2 (eg) θ 2 (eg/s) velocity (m/s) velocity (m/s) Stance Phase Swing Phase The goal of the bon graph moel is to obtain the power flow in the bons that is representative for the Linkage Leg mechanism. The final parameters of the Linkage Leg are not know at this point, so three test cases will be simulate. In orer to simulate the leg using the bon graph shown in Figure 3.44 the geometric parameters in Table 3.1 are use, together with the masses an 27

37 moments of inertia from Table 3.2. The bon graph is initialize using the initial conitions in Table 3.3. During the swing phase the knee is free to rotate, an uring the stance phase a 38.4Nm/eg spring is applie aroun the knee joint. Figure 3.5 shows the trajectories of the hip an knee joints uring one step. The takeoff an touchown points are also inicate. The bon graph has been simulate for a representative step motion, with the robot running at 3m/s an reaching 1.5m apex height, similar to a human gait. The swing an phase moels are simulate inepenently, causing the instantaneous change in velocity at touchown. The groun moel is use uring the stance phase, with a N/m spring 1.5 Takeoff Touchown Takeoff vertical height (m) hip angle (eg) horizontal istance (m) takeoff angle touchown angle hip angle Takeoff Swing time (s) Touchown Takeoff Stance knee angle (eg) Takeoff Touchown Takeoff time (s) Figure 3.5 Trajectories of the knee an hip joint of the bon graph moel. 28

38 an 100 N/(m/s) amper holing the toe point locke to the groun. The spring an ampers are set to zero uring the swing phase so the boy is free to move accoring to the projectile motion. The stance an swing phases are simulate inepenently, so the more complicate groun mechanism [63] use in Chapter 2 is not necessary. This section etaile the moelling of the Linkage Leg using bon graphs. The next section will explain how the bon graphs moel is simplifie to obtain the AS mechanisms Bon graph moel simplification Simplification algorithms This section efines a metho of etermining the analogous simpler mechanisms for a given mechanism using its bon graph moel, an applies it to the Linkage Leg. The propose metho is a two-step process. The first step etermines the ynamic elements that are necessary for the behaviour to remain similar to that of the original system, an the secon step eliminates the bons that have insignificant power flow an etermines what mechanism is forme by the remaining ynamic elements. The term ynamic element refers to the bon graph component associate with masses an moments of inertia, generally represente by inuction components. The first step, calle Dynamic Element Simplification (DES), is an element-oriente simplification approach that eliminates the maximum number of ynamic elements while keeping the ynamic behaviour unchange [67]. The algorithm requires a set of ynamic elements S to be simplifie. If there are m ynamic elements in the bon graph (such as masses an moments of m inertia) an simplification is to be one for n of them, then there are C m n possible combinations. For each possible combination, the bon graph is simulate, an the behaviour of a set of test bons is compare, such as the bons relate to the controlle egrees of freeom of the system. The result of this first step is a set of elements that prouce the closest behaviour to the original system for each level of simplification, as measure by Root Mean Square (RMS) error [67]. The level of simplification represents the number of remove ynamic elements (level zero for no simplification, level one for one element remove, level two for two elements remove, etc.). Beyon a certain simplification level, the system behaviour will start markely eviating from the original behaviour, inicating that all the remaining elements are necessary. This effectively sparsifies the ynamic elements of the bon graph. The secon simplification step, the Iniviual Bon Simplification (IBS), can be applie to any of the simplification levels from the first step. The goal is to fin the necessary bons for the 29

39 system behaviour to remain relatively unchange. First, a metric is foun that represents the importance of each bon an component for the bon graph. Bons with zero or small power flow get reuce. The metric use in this step is base on the correlation of the energy flow pattern in the bons of the bon graph. The en result is a rank for each bon that represents the importance of that bon to the overall system behaviour. The algorithm, base on the approach presente in [66], is summarize below: a) Fin energy trajectories for each bon (E i ) from Fig. 3.5, an form the E matrix E = [E 1, E 2... E n ]. The algorithm can analyze the entire bon graph or only a selection of bons. The energy trajectories are obtaine by simulating the bon graph for the esire behaviour. b) Apply Singular Value Decomposition (SVD) to the E matrix, an calculate the U, S, V matrices such that E = USV 1. n i=1 c) Calculate vector I = σ 2 i v i, where n is the number of bons analyze, σ i=1..n are the elements of the iagonal matrix S, an v i is the i th column of vector V. ) Normalize the vector I an orer the bons in the orer of ecreasing relative importance. e) Choose a value r to be the cut-off threshol. Every time I i+1 I i < r, a new reuction stage is reache. The cutoff threshol is chosen iteratively to result in clearly etermine stages that correspon to meaningful parts of the bon graph [66]. For this analysis r = 1.5. f) Finally, one can choose as many reuction stages as necessary. The first two or three reuction stages usually contain the meaningful ynamics. The rationale for the propose approach consisting of both the DES an IBS steps is as follows: most methos, such as those presente in [66], [79] an [80] which form the basis for the IBS, are base on the notion of eliminating the bons with zero or small power flow. For some cases, these methos are able to fin the unerlying simple mechanisms [79]. However, these methos are not suitable as a complete solution for mechanisms with parallel links, for example the Linkage Leg with the tibia-tenon an thigh-foot pairs, as it will be iscusse in sequel. Therefore, if only the secon simplification step is applie to the Linkage Leg, all the links will be consiere as important an consequently no mass will be eliminate in the simplification process. The DES step is necessary for ealing with not only the insignificant iniviual masses but also the parallel links of the Linkage Leg by fining the minimal combination of masses that maintains the original system ynamics, regarless of power flow. Once the minimal set of significant masses 30

40 is foun, IBS is able to etermine the bons that are significant in transmitting the energy in the step-1 mechanism. It shoul be note that the propose simplification approach is epenent on the system motion phase as well as the initial conitions for each phase. In other wors, the simplification shoul be one locally for a certain system behaviour, an it oes not cover the entire ynamics of the system, because the significance of ifferent ynamic elements varies in ifferent system behaviours. Therefore, one must first efine the working conition of the system with its motion phases an initial conitions prior to performing moel simplification. This will be further iscusse in the following section Linkage Leg moel simplification To etermine the working behaviour of the Linkage Leg two simulations were initially performe for the two motion phases. The trajectories of the hip an the knee joints were consequently obtaine, an they are shown in Figure 3.6, for the stance phase, an Figure 3.7, for the swing phase. Also, the proper initial conitions for both phases were obtaine, as shown in Table 3.3. The following moel simplifications will be base on the system behaviour in stance an swing phases, which resulte from the obtaine initial conitions. This local simplification approach woul provie more effective results that are irectly relevant to the intereste working behaviour of the system. For many systems, a global approach of simplification that can result in a unique simpler system resembling the original one at all working conitions may be inefficient or infeasible, as it is argue in [79]. To simplify the Linkage Leg bon graph moel, three cases are investigate. The test cases are meant to reflect ifferent construction possibilities for the Linkage Leg, base on CAD moels of possible configurations. These cases, liste in Table 3.2, were chosen to represent ifferent construction methos an leg configurations. The first one represents a more natural leg, with the thigh (m1) an tibia (m2) heavier than foot (m3) an tenon (m4). The thir case represents a more classic robot construction, with all the linkages of equal mass, with only the tenon being lighter. The secon case is in between the first an thir ones, with the thigh heavier than the tibia an foot, which are equal, an the tenon lighter. In all three cases the Linkage Leg geometry is kept constant as shown in Table 3.1 an the mass an moment of inertia parameters are change, as liste in Table

41 The first step of the simplification (DES) requires a set of ynamic elements to be simplifie an the bons use to test the similarity to the original system. Similar to most robotic legs, the Linkage Leg has two egrees of freeom, the hip (θ 1 ) an the knee (θ 2 ) joints. These two joints are also the actuate an controlle joints. Therefore, the bons associate with the moments of inertia of these two joints are use to test the similarity of each simplifie combination to the original system, as explaine in [67]. The components associate with the other ynamic elements (masses m 1, m 2, m 3, m 4 of the linkage leg boies, the mass m B of the robot boy an moments of inertia I 3 an I 4 ) form the set of elements to be simplifie. The results of the first simplification step, presente in Table 3.4, show the combination of elements with the lowest cost (closest fit to the original ata) for each simplification level of the Linkage Leg in figure 2.1. The cost function is the square ifference between the trajectories of the test joints (O 1 an O 2 ) of the simplifie simulation an those of the original case. These trajectories for case 3 are shown in Figure 3.6, for the swing phase, an in Figure 3.7, for the stance phase. The unmoifie trajectory is the same as the one in Figure 3.4. Table 3.4 Results of the ynamic element simplification (DES) step. Simp. Level SWING PHASE STANCE PHASE Case 1 Case 1 m1 m2 m3 I3 m4 I4 mboy cost m1 m2 m3 I3 m4 I4 mboy cost Level Level Level Level Level Level Level Case 2 Case 2 m1 m2 m3 I3 m4 I4 mboy cost m1 m2 m3 I3 m4 I4 mboy cost Level Level Level Level Level Level Level Case 3 Case 3 m1 m2 m3 I3 m4 I4 mboy cost m1 m2 m3 I3 m4 I4 mboy cost Level Level Level Level Level Level Level

42 hip angle (eg) Takeoff Touchown knee angle (eg) time (s) The simplification levels start from level zero, with no simplifie elements, an go up to level six, where all the ynamic elements of the linkage leg are remove. The ata shows that the mass of the robot boy is the most important element in all cases, an was never simplifie. For the swing phase, the lower simplification levels are ifferent between the test cases, inicating that the least important ynamic elements vary epening on the test case being consiere. However, the higher simplification levels are the same for all test cases. If four elements are simplifie an two are kept intact, m 1 an m 3 are etermine to be the important elements, an if only one element is kept, m 2 is etermine to be the most influential. In Figure 3.6 it can be seen that simplification levels five an six have markely ifferent trajectories for the joints on interest, O 0 an O 1, while simplification level four shows a closer behaviour to the unmoifie system. Consequently, the simplification levels four an five will be analyze further. It shoul be note that to obtain a vali comparison of hip an joint trajectories between ifferent simplification levels in Figure 3.6, the hip torque τ hip applie uring the swing phase must be proportional with the overall moment of inertia of the leg aroun the hip joint. For the stance phase, Figure 3.7, the Linkage Leg behaviour remains relatively unchange no matter how many ynamic elements in the leg mechanisms are simplifie. This is explaine by the much larger mass assigne to the boy of the robot an the fact that the toe is in contact with the -118 time (s) Figure 3.6 Swing phase joint angle results for DES step, Case 3. It can be seen that the last two simplification levels, 5 an 6, iffer greatly from the original behaviour. As such, simplification level 4 is the one use in the IBS step. 33

43 hip angle (eg) Touchown Takeoff knee angle (eg) No Simplification Simp. level 1 Simp. level 2 Simp. level 3 Simp. level 4 Simp. level 5 Simp. level time (s) time (s) Figure 3.7 Stance phase joint angle results for DES step, Case 3. It can be seen that all simplification levels have similar behaviour, so all the ynamic elements of the leg can be eliminate an the behaviour will be close to that of the original leg. groun, which ominates the ynamic behaviour of the joints O 1 an O 2. The case of all the elements being simplifie is the one with the most ifferent behaviour, as expecte. The next step is to unerstan what kin of mechanism is escribe by the remaining ynamic elements. For the stance phase, the combination resulting from the sixth simplification level is investigate (all ynamic elements of the leg, but not the robot boy, are remove), whereas for the swing phase the combinations from resulting from the fourth an fifth simplification levels are investigate. Given that the results are very similar for all three initial conitions cases, for the sake of brevity they will be presente only for case 3. The results of the secon simplification step are shown in Figure 3.8 an Figure 3.9 for the swing phase, an, in Figure 3.10 for the stance phase. The meaningful bons are shown in black, an the reuce bons are graye out. For the fifth level of simplification, shown in Figure 3.9, the gravity an inertial effects of m 2 remain unchange, together with the vector efining the location of the center of mass relative to O 1. The tibia is linke to the robot boy by the massless ro {O 0 O 1 }, represente in the bon graph by the remaining elements of the boy {O 0 O 1 O 4 }, forming a ouble penulum system. For the simplification level four (Figure 3.8), boy m 1 is left 34

44 Figure 3.8 Swing phase, simplification level four Figure 3.9 Swing phase, simplification level five 35

45 Knee Spring Groun Contact Moel Figure 3.10 Stance phase, simplification level six intact, but the bons associate with the mass m 2 are remove. Instea the mass m 3 acts at the point O 2, effectively becoming part of the boy {O 1 O 2 }. This also forms a ouble penulum mechanism, this time however both links have mass an inertia. This ifference explains the change in behaviour between simplification levels four an five, inicating that a ouble penulum where both links have mass is the simplest mechanism that has similar behaviour to the original linkage leg. The bons associate with the position of the mass m 3 relative to point O 2 have an effect on the system behaviour, moving it away from a perfect ouble penulum similarity. The effect is iscusse below, in the comparison section. The stance phase bon graph for simplification level six is shown in Figure The robot boy ynamics remain unchange, an the other remaining elements are the groun contact moel an the spring of the joint O 1. The rest of the bons are kinematic bons, which escribe the location of the boy with respect to the groun contact point O 3, as well as the orientation of the boy velocity with respect to the groun. 36

46 The resulting mechanism is similar to the SLIP moel, wiely use for running an hopping robots. The main ifference is that in the SLIP moel the irection of the force ue to the spring is coincient with the hip joint [20], whereas in the Linkage Leg this irection is etermine by the instantaneous configuration of the four bar linkage system [81], an changes epening on the leg length Comparison between Linkage Leg an the simple mechanisms The ouble penulum is propose as a simpler mechanism caniate for the swing phase an the SLIP moel as a caniate for the stance phase. As inicate in the bon graph, the first rigi hip angle (eg) knee angle (eg) Simp. level 4 Equivalent ouble penulum hip angle (eg) knee angle (eg) -115 time (s) time (s) Figure 3.11 Comparison between Linkage Leg an ouble penulum Simp. level 4 Equivalent ouble penulum time (s) time (s) Boy Position X(m) Boy Position Y(m) time (s) time (s) Figure 3.12 Comparison between Linkage Leg an SLIP moel 37

47 boy of the ouble penulum is compose of the mass m 1 an rotational inertia I 1, an the secon rigi boy is compose of the mass m 3 an rotational inertia I 2. The comparison between the Linkage Leg an the ouble penulum is shown in Figure 3.11, an it shows that the behaviour of the hip an knee trajectories is similar. The hip trajectory of the ouble penulum has a 10% maximum ifference compare with the linkage leg, while the knee is closer with a 5% maximum ifference, as measure by RMS. This ifference between the ouble penulum an linkage leg is explaine by the effect of the moment arm bons on the m 3 mass. This effect can be minimize epening on the location of mass m 3 relative to the point O 2. A mass offset x 3 of 0.5 results in the mass m 3 coincient with O 2, whereas an offset of 0.35 results in the mass m 3 in the center of the segment O 3 O 5. Both the Linkage Leg an the ouble penulum trajectories en up at the same touchown positions. The leg angles at touchown are shown to have the greatest impact on the running gait in [20], further inicating suitability of the ouble penulum for this particular application. The SLIP moel is compose of a massless spring between the boy center of mass an the toe contact point [20]. Given that the SLIP moel oes not have the same hip an knee joints of the Linkage Leg, the two systems are compare base on the position of the center of mass of the robot boy. The comparison is shown in Figure Both the horizontal an the vertical position of the boy are very similar between the Linkage Leg an the SLIP moel, inicating that the offset between the hip an the spring force present in the Linkage Leg has little influence on the final behaviour. The maximum ifference of 0.02m between boy positions is similar to the ifference foun between the moels shown in [66] Bon graph stage conclusions A metho was presente that fins simpler mechanisms with a ynamic behaviour similar to that of a more complicate robotic leg mechanism for each specific motion phase. Such simpler mechanisms will be goo representatives of the original mechanism while having the avantages of simplicity an tractability. The original leg mechanism was first moele using a bong graph approach, an then a two-step simplification process was use to etermine simpler mechanisms with the same significant ynamic characteristics. The first step fins the ynamic elements that are necessary for the system behaviour to remain unchange for each motion phase, an the secon step simplifies the bon graph forme by such elements an constructs the resulting mechanism expresse by the simplifie graph. 38

48 For the case stuy of a Linkage Leg two motion phases were analyze, i.e., swing an stance. During the swing phase, two masses elements were foun to be require for generating the same behaviour as the original system, resulting in a ouble penulum mechanism. An, for the stance phase the resulting simplifie mechanism was foun to be the classic SLIP moel that has been use for moeling robotic an biologic leg ynamics. Having such simpler mechanisms will enable the esigner to fine-tune the kinematic, ynamic an control parameters of the original mechanism in a more intuitive way using methos such as the one iscusse in the following section. 3.3 Analogous System concurrent optimization The goal of the AS Concurrent Optimization stage is to obtain the best AS mechanism an controller for each respective motion phase. Since the AS is compose of ifferent mechanisms, each representing a vali analogy to the ES for its corresponing motion phase controllers are esigne for each AS mechanism only uring their respective motion phase. Once the transition conitions have been ientifie an control strategy chosen, each AS mechanism an control parameters can be concurrently optimize. The first step in the AS Concurrent Optimization stage, Figure 3.13, is to efine the control strategy for the AS epicte in Figure The swing state has two controllable egrees of freeom. They are chosen to be the angle of the leg with respect to the groun, θ L, controlle by the torque at the hip (τ 1 ), an the angle of the knee, θ 2, controlle by the torque at the knee (τ 2 ). The stance phase has one controllable egree of freeom, chosen to be the angle of the boy with respect to the groun, θ B, controlle by the torque at the hip. Furthermore, uring the stance phase Figure 3.13 Diagram of the Analogous System Concurrent Optimization phase 39

49 a constant force (f L ) is applie to the leg joint when the leg is extening, helping to offset any energy loss uring the gait [20]. x st,i = sw st (x sw,f ) Swing Phase x sw = f sw (x sw, τ 1,sw, τ 2,sw ) Stance Phase x st = f st (x st, τ 1,st, f L ) x sw,i = st sw (x sw,i ) x sw,i The selecte control strategy consists of a esire trajectory efine by a Bezier curve [25] for each controllable egree of freeom, an a PD controller that attempts to follow the esire trajectory. Bezier curves were chosen as they allow for a simple way to efine beginning an en points an guarantee smoothness. The formula for the Bezier curve is: M j M j! Figure 3.14 Analogous System θ j (s) = (P j ) k k!(m j k)! sk (1 s) M j k k=0, j {θ L, θ 2, θ B }, s [0,1] (3.1) 40

50 where the superscript enotes the esire values. Each Bezier curve in (3.1) is efine by the number of control points, M j, an by the vector of control points, P j. For the hip joint uring swing an stance M θl = 4 an M θb = 4, respectively. One of the useful properties of Bezier curves is that the first an last control points represent the initial an final trajectory values, an the secon an secon-last control points etermine the initial an final slope of the trajectory. Consequently, the hip joint the control point vectors are: P θl = [θ L,sw,i θ L,sw,i + θ L,sw,i θ L,sw,f P θb = [θ B,st,i θ B,st,i + θ B,st,i θ B,st,f + θ L,sw,f + θ B,st,f θ L,sw,f ] (3.2) θ B,st,f ] (3.3) The subscript sw enotes the swing phase, the subscript st enotes the stance phase, an the subscripts i an f refer to the initial an final values of the variables in their respective motion phase. The knee trajectory for the swing phase has one intermeiate control points, so the knee can be retracte. The initial an final position of the knee trajectory are the same. P θ2 = [θ 2,sw,i θ 2,sw,i + θ 2,sw,i θ 2,inter The vector of esire parameters α = [θ L,sw,f θ 2,sw,f, θ L,sw,f + θ 2,sw,i, θ 2,sw,f, θ 2,inter θ 2,sw,i ] (3.4), θ B,st,f, θ B,st,f, f L. ] T efines the seven tunnable parameters of this control system. The esire touchown angle an angular velocities of the leg are efine as θ L,st,f an θ L,st,f, which affects the hip trajectory uring swing. The knee trajectory uring the swing phase is controlle by the knee touchown velocity, an the intermeiate Bezier control point θ θ 2,st,f 2,st,f. During the stance phase the hip trajectory is controlle by the esire boy angle an angular velocity at takeoff (θ B,st,f an θ B,st,f ) an the force in the leg, f L. The rest of the Bezier control points from Eq cannot be irectly specifie, but are given by the actual configuration of the robot at the start of the respective motion phase. The PD controller is efine by the proportional gain K P an ifferential gain K D. A servo control strategy was chosen because of its simplicity, given that the goal of the present work is not to etermine the best control strategy for a running robot, but to show that, given a selecte control strategy, MbA offers a better alternative to the traitional esign optimization proceures. During the stance phase the knee joint is actuate only when the keg is extening, using a constant force, f L,st, similar to the strategy use in [20]. 41

51 The next step of the AS Concurrent Optimization stage is to unerstan the transition conitions between the phases. The configuration space for the swing phase is parameterize using the Cartesian position of the hip joint (x 0, y 0 ), the angle of the robot s boy with respect to the groun (θ B ), as well as the angles of the hip (θ 1 ) an knee (θ 2 ), i.e., q sw = [x 0,sw y 0,sw θ B,sw θ 1,sw θ 2,sw ] T, x sw = [q sw q sw] T. The parameterization for the stance phase is one using the egrees of freeom of the SLIP moel: the angle of the leg with respect to the groun (θ L ), the length of the leg (l), an the hip angle (θ H ), as shown in Figure 3.14, i.e., q st = [θ L,st l st θ H,st ] T, x st = [q st q st] T. At the touchown an takeoff transitions the parameterization must be change from the swing to the stance phase, an from stance to swing, respectively. The initial state of the stance phase, x st,i, is efine as a function of the final state of the swing phase, namely x st,i = sw st (x sw,f ). Similarly, the initial state of the swing phase, x sw,i, is efine as a function of the final state of the stance phase, i.e., x sw,i = st sw (x st,f ). tan 1 (y 0,sw,f (x 0,sw,f x toe,sw,f )) (x 0,sw,f x toe,sw,f ) 2 + y2 0,sw,f sw st = π φ offset θ 1,sw,f ((x 0,sw,f x toe,sw,f )y 0,sw,f y x 0,sw,f 0,sw,f) ((x 0,sw,f x toe,sw,f ) y 0,sw,f ) (3.5) ((x 0,sw,f x toe,sw,f )x 0,sw,f + y 0,sw,f y ) 0,sw,f (x 0,sw,f x toe,sw,f ) 2 + y2 0,sw,f [ θ 1,sw,f ] st sw = l st,f cos θ L,st,f l st,f sin θ L,st,f θ L,st,f + θ H,st,f π φ offset θ H,st,f (l st,f l st,i )C l st,f cos θ L,st,f l st,f sin θ L,st,f θ L,st,f l st,f sin θ st,l + l st,f cos θ L,st,f θ L,st,f (3.6) θ L,st,f + θ H,st,f θ H,st,f [ l st,fc ] 42

52 The parameter C in (7) represents the coupling between the rate of change of leg length, l, an the knee joint, θ 2, of a Linkage Leg with the same geometry as that efine by the ouble penulum, as explaine in section 3.3. To ensure that the potential energy remains the same while transitioning between the AS mechanisms, the SLIP masses, moments of inertia an locations of the centre of mass must be equal to those of the ouble penulum, such that the following equations are satisfie: m s = m 1 m U = m 2 I S = I 1 I U = I 2 x s = ((x 1 + a 1 ) cos( offset ) y 1 sin( offset )) y s = ((x 1 + a 1 ) sin( offset ) + y 1 cos( offset )) x U = l sw,i (a 1 cos( offset ) + (a 2 + x 2 ) cos( offset + 2,sw,i )) y 2 sin( offset + 2,sw,i ) y U = (a 1 sin( offset ) + (a 2 + x 2 ) sin( offset + 2,sw,i )) + y 2 cos( offset + 2,sw,i ) The thir stage is to optimize the AS mechanical system, Bezier trajectories an controller gains concurrently for the stance an swing phases. The optimization is performe for one gait cycle, consisting of swing an stance phases, as shown in Figure (3.7) The ynamics of the ouble penulum an robot boy uring the swing phase are expresse in the equation (3.8): f sw = [ M sw 1 x sw = f sw (x sw, τ 1, τ 2 ) q sw C sw (q sw, q sw )q sw g sw (q sw ) τ 1 ( [ τ 2 ] )] (3.8) The matrix M sw is the mass matrix of the ouble penulum an robot boy mechanism, the matrix C sw is relate to the Coriolis an centrifugal effects, an g sw is the column vector expressing the effects of gravity. The values in the torque column vector are τ 1, the torque in the hip joint, an τ 2, the torque in the knee joint. The ynamics of the SLIP leg an robot boy system uring the stance phase are expresse in a similar fashion in the equation (3.9): 43

53 x st = f st (x st, τ H, f L ) q sw 0 f st = M 1 st ([ f L ] C st (q st, q st )q st g st (q st [ τ H ))] (3.9) The equations (3.8) an (3.9) together with the transition functions (3.5) an (3.6) efine all the equations necessary to simulate the behaviour of the AS system uring a gait cycle. The variables to be optimize are shown in Table 3.5, an they are ivie into three categories: the AS mechanical esign variables, efining the imensions, mass an inertia parameters of the AS, the Bezier trajectory variables, efining the esire trajectories of θ L an θ 2 for the swing phase an θ B for the stance phase, the control gains, as well as the variables efining the takeoff state at which the simulation is initialize (x sw,i ). Table 3.5 Analogous System Optimization variables Physical Variables Boy Parameters variable units Initial value Lower boun Upper boun Optimize value m B kg I B kg m x B m y B m Double Penulum Parameters variable units Initial value Lower boun Upper boun Optimize value m 1 kg I 1 kg m x 1 m y 1 m m 2 kg I 2 kg m x 2 m y 2 m a 1 m a 2 m SLIP Parameters are require to be equal to the ouble penulum parameters variable units Initial value Lower boun Upper boun Optimize value m S kg I S kg m x S m y S m m U kg I U kg m x U m y U m l st,i m

54 φ offset eg K SLIP kn/m Desire Trajectory Parameters (α) variable units Initial value Lower boun Upper boun Optimize value eg θ L,sw,f θ L,sw,f θ 2,sw,f θ 2,inter eg/s eg/s eg θ B,st,f eg θ B,st,f eg/s f L N Initialization state (x sw,i ) variable units Initial value Lower boun Upper boun Optimize value θ L,sw,i eg θ L,sw,i eg/s L sw,i m L sw,i m/s θ B,st,i eg θ B,st,i eg/s Control Parameters K P Nm/eg K D Nm s/eg The optimization constraints, shown in Table 3.6, are require such that the behaviour of the system represents a proper running gait. The most important characteristic of a running gait is its repeatability [20]. This is represente by the equality constraints 1-6 which require the linear an rotational positions an velocities of the robot boy to be consistent between the initial state of the swing phase an the final state of the stance phase. Table 3.6 Analogous System Optimization constraints Equality Optimization Constraints # θ B,sw,i θ B,st,f = 0 Difference between the initial an final hip angle 1 θ B,sw,i θ B,st,f = 0 Difference between the initial an final hip angle 2 x 0,sw,i x 0,st,f = 0 Difference between the initial an final hip horizontal position 3 y 0,sw,i y 0,st,f = 0 Difference between the initial an final hip height 4 x 0,sw,i x 0,st,f = 0 Difference between the initial an final hip horizontal velocity 5 y 0,sw,i y 0,st,f = 0 Difference between the initial an final hip vertical velocity 6 (m 1 + m 2 ) (m U + m S ) Mass of AS mechanism is the same in both phases 7 I sw I st = 0 Moment of inertia of the stance AS mechanism is equal to that 8 of the swing AS mechanism x com,sw x com,st = 0 Difference between the horizontal location of the centre of mass of the AS mechanisms is zero 9 45

55 y com,sw y com,st = 0 Difference between the vertical location of the centre of mass 10 of the AS mechanisms is zero Inequality Optimization Constraints y 0,sw,i > 0 Vertical velocity at takeoff is positive 11 min(y 0 ) > 0 Minimum height of the hip centre of mass is positive 12 min(x 0) > 1 Forwar velocity is above 1m/s 13 max (τ 1 ) < 50 max (τ 2 ) < 50 Maximum torque is below 50Nm 14 max (τ H ) < 50 where: I sw = m 1 ((a 1 + x 1 ) 2 + y 1 2 ) + m 2 (a a 1 y 2 + (a 2 + x 2 ) 2 + y a 1 cos θ 2 (a 2 + x 2 ) 2a 1 y 2 sin θ 2 ) + I 1 + I 2 is the moment of inertia of the ouble penulum (swing phase AS mechanism) aroun the O 1 joint, an: I st = I U + I S + m S (x S 2 + y S 2 ) + m U ((l st + x U ) 2 + y U 2 ) is the moment of inertia of the SLIP (stance phase AS mechanism). The inequality constraints are use to make sure the gait is esirable for a legge robot. Constraint 11 requires that the vertical velocity at takeoff be positive, so that the robot can leave the groun. Constraint 12 requires that the velocity always stay above the esire value of 1m/s, an constraint 13 requires that the minimum height of the hip be positive. The last constraint requires that the maximum torque values of the hip an knee joints be below 50Nm, as a typical upper boun for motors use for legge robots [68]. The cost function of the AS optimization is 1/ max(x 0). Minimizing this cost function results in the highest possible forwar spee given the optimization constraints. Notably, the maximum torque constraint is limiting the achievable maximum spee. Three gait cycles of the optimize AS are shown in Figure 3.15, an the behaviour of the initial an optimize AS are compare in Figure The horizontal an vertical velocities at the beginning of the swing an the en of the stance are not equal, so the initial AS, shown with otte lines, oes not achieve a repeatable gait with the initial control vector α an initial configuration Figure 3.15 Three steps of the optimal AS system 46

56 Figure 3.16 Comparison of the behaviours of the initial AS system (otte line) an the optimize AS system (soli line) for one step. x sw,i.the optimize AS, shown using soli lines, achieves both a repeatable gait as well as a faster spee than the initial AS. The optimal hip an knee toque require by the optimal trajectories is much lower than the torque of the initial case. The optimize behaviour is similar to the one shown by other single legge hopping robots [20]. The velocity of the hip is fairly constant uring the swing phase, an ecreases in the mile of the stance phase [55]. The touchown angle of the leg was foun to be the most important factor affecting the forwar velocity of the robot [55], an the optimization proceure was shown to result in a trajectory of the hip, θ L, which ensures a touchown angle that maintains constant forwar velocity. The hip trajectory uring stance phase, θ B, ensures that the boy angle at the en 47

57 of stance is the same as the angle at the beginning of swing, so the gait cycle is repeatable as seen in Figure Robust trajectory control The controller escribe previously follows esire trajectories that, if optimize, can lea to a repeatable gait, but is not able to eal with isturbances that affect the unactuate egrees of freeom. The angular isplacement of the boy an its centre of mass velocity uring the swing phase are especially ifficult to control, since the robot has no contact with the groun. Hence, a top-level control is require to help the robot eal with isturbances. Two approaches exist in the literature to achieve this goal. The first is to moify the simple PD controller so that it can account for exteroceptive errors [82], an the secon approach is to moify the previously obtaine optimal trajectories of subsequent gait steps to account for the errors in the current step. The first approach is a continuous-time control, an the secon one is iscrete time control, upating the controller only once for each step of the gait, an it is thus more esirable for lowering the require computational power of the system. This approach of controlling the joint trajectories base on the error of a Poincare point is common in the control of walking an running robots [18]. The top-level control consists of a control matrix K that relates the error in the states of the robot at a certain time in the gait to the change in the trajectory parameters in the subsequent step. The instant at which this controller is applie (Poincare point) was chosen to be the takeoff moment. The states of the robot at takeoff are efine by three variables, x st,f = [θ L,st,f, θ L,st,f, l st,f, l st,f, θ B,st,f θ B,st,f ] T an their erivatives, similarly to the way the stance states are efine. The top-level control is efine in Eq. (3.10): Δα = K (x st,f x st,f ) (3.10) The control gain K is a 7-by-6 matrix that calculates each of the esire parameters in α, epening on the errors between the takeoff states x st,f an the esire takeoff states x st,f. The task now becomes to esign the matrix K in such a way that the vector of esire parameters α o not eviate too much from that obtaine in the AS concurrent optimization, but is also able to eal with external stimuli. Several of the takeoff states of the robot, such as the angular velocity θ B,st,f an the leg extension rate, l st,f, cannot eviate too much from their nominal values, or the subsequent step will fail with no chance for recovery. 48

58 In [20] it is assume that the horizontal an vertical components of the velocity of the boy centre of mass at takeoff can be ecouple, with the former being controlle by the leg touchown angle (θ L,sw,f ) an the later by the force in the leg uring stance (f L ). If this strategy was use, the matrix K woul have only two non-zero elements, (1,4) an (7,5). However, a better performance can be obtaine if all the elements of α are controlle epening on the takeoff state, an thus have a full matrix K [18]. Figure 3.17 AS two-layer control strategy Discrete-time Linear Quaratic Regulator (LQR) is a classic solution to the problem of esigning the controller in Eq. (3.10). The LQR takes a iscrete-time linear system x k+1 = A x k + k=0 Bu k an a performance inex J = (x T k Qx k + u T K Ru k ), an calculates the optimal gain K that minimizes the eviations of the states an control inputs from their esire values [83]. Matrices Q an R efine the weight associate with the states an the inputs, respectively. The esigner must supply these matrices, which are often foun by trial-an-error. To obtain the A an B matrices escribing the system, the system in Figure 3.14 must be linearize. A step of the AS is forme by a swing phase (f sw ), a swing-to-stance transition ( sw st (x sw,f )) an a stance phase (f st ). The Poincare return map function x st,f,k+1 = 49

59 f AS (x st,f,k, α k ) is constructe to represent this step, where x st,f,k represents the takeoff state at the beginning of the current step, α k represents the esire trajectory parameters, an x st,f,k+1 is the takeoff at the en of the current step. This function has no close-form solution, but has to be numerically evaluate by simulating the step. The linear system neee for the LQR controller is achieve by taking the Jacobians of the f AS function, as seen in Eq. (3.11). x st,f,k+1 = f AS (x sw,i,k, α k ) nonlinear system A = f AS(x sw,i,α) x sw,i,k Jacobian with respect to states B = f AS(x sw,i,α) Jacobian with respect to input α x sw,i,k+1 = Ax sw,i,k + Bα k linearize system (3.11) For the values optimal AS shown in table 3.5, the A an B matrices are: A = f AS(x sw,i, α) x sw,i,k = B = f AS(x sw,i, α) α [ ] = [ ] Fining the Q an R matrices usually involves trial-an-error until the values resulting in a stable AS gait over more than 30 steps. The values of the Q an R matrices, together with the values obtaine for A an B for the AS escribe in Table 3.5 are shown in below K = [ ] 50

60 Q = , R = [ ] [ ] It was foun that the values corresponing to the ifferential of the state values in x sw,i (rows 1, 3 an 5 of matrix Q) neee to be much higher than the rest. For the last row of the matrix R, corresponing to the cost of the leg force, the value neee to be finely ajuste so that the gait oes not exhibit a tall-short, ouble-step pattern similar to the one iscusse in [20]. The matrix B is quite informative as to the behaviour of the AS mechanism. The forwar velocity (row 2) an vertical velocity (row 4) at takeoff can be controlle very efficiently by a small change in the touchown angle (column 1). The last column, representing the effect of the leg force, also affects these two states. The takeoff boy angle an angular velocity can be irectly controlle by ajusting θ B,st,f an θ B,st,f, respectively, since B(5,5) an B(6,6) are both very close to 1. However, the vertical an horizontal components of the boy centre of mass cannot be controlle inepenently, so an LQR control is more suitable for this case that the metho suggeste in [20]. The overall control strategy employe for the AS can be summarize in Fig The lowlevel control is a PD continuous-time control which ensures that the esire trajectories are followe. The top-level control etermines the esire trajectory parameters for the subsequent step base on the ifference between the actual an esire takeoff states. One of the avantages of the MbA methoology is that the controller is base on the simpler AS, so that the A an B matrices can be easily unerstoo. This leas to a more transparent an intuitive control esign. 51

61 3.4 Dimensional analysis stage The goal of the Dimensional Analysis stage is to etermine the parameters of the ES (the Linkage Leg) which provie the closest behaviour to that of the optimize AS (ouble penulum an SLIP). To achieve this goal the concept of ynamic similarity as iscusse in [70], is applie to the imensionless Lagrangian equations of the ES an AS. If the similarity is achieve, it means that for any given state vector q LL the potential an kinetic energies of the Linkage Leg system are equal to those of the AS in either motion phase. Because this similarity hols for any state space vector, it also means that the same change in state space vectors prouces equal changes in the energies of the system. While similarity requires exact match between the two systems, in the imensionless analysis step it was foun that an approximate similarity (here calle analogy) is sufficient to achieve the esire goals of the MbA methoology. The approximate conitions result in systems with close ynamic behaviours, as it will be shown in the results section. The steps of the Dimensional Analysis stage are shown in Figure First, the imensionless Lagrangian equations of the mechanisms forming the AS are obtaine, an the Pi-factors are calculate for the AS. Seconly, the Pi-factors are obtaine from the imensionless Lagrangian of the Linkage Leg. Finally, an optimization proceure is use to fin the parameters of the Linkage Leg which result in the closest Pi-factors between the two systems. The subscript AS has been use in the following equations to ifferentiate the variables of to the AS from those of the Linkage Leg, for which the subscript LL will be use. Figure 3.18 Diagram of Dimensional Analysis stage 52

62 The imensionless Lagrangian equation of the ouble penulum leg attache to a robot boy can be expresse as: L AS,sw = ( (q T AS,sw) ) V T AS,sw ( (q AS,sw) ) + G AS T AS,sw Q AS,sw (3.12) AS The mass matrix V AS,sw, pre- an post-multiplie by the time erivative of the state vector q AS,sw, efines the kinetic energy of the system. The term G AS,sw Q AS,sw efines the potential energy of the system, where Q AS,sw is a convenient combination of the cosine an sine of the elements of q AS,sw. The V AS,sw an G AS,sw matrices are efine in the appenix. Notably, the time an energy units are change to T AS = a AS,1 g t an L AS,sw = L AS,sw (I AS,2 + m AS,2 (y AS,2 2 + (a AS,2 + x AS,2 ) 2 ) ) g a AS,1, respectively, following the proceure outline in [84], which allows the comparison of imensionless equations of motion. The Lagrangian equation for the SLIP moel shown Figure 3.14 is: L AS,st = ( (q T AS,st) ) V T AS,st ( (q AS,st) ) + G AS T AS,st Q AS,st + 1 (l AS 2 st l st,i ) 2 K AS (3.13) where K AS is the spring coefficient of the SLIP mechanism, an l st,i is the leg length at touchown. The formulas for V AS an G AS matrices for both swing an stance phases are also inclue in the appenix at the en of the section. The Lagrangian of the Linkage Leg uring the swing phase is: L LL,sw = ( δ(q T LL,sw) δ(t LL ) ) V LL,sw ( δ(q LL,sw) δ(t LL ) The expressions for V LL,sw an G LL,sw can be foun in the Appenix. ) + G LL,sw Q LL,sw (3.14) The stance phase as two constraints to the ynamics of the Linkage Leg, namely that the vertical an horizontal positions of the toe be zero. To obtain the Lagrangian for the stance phase q LL,sw an Q LL,sw have to be converte to a new set of coorinates that incorporate these kinematic constraints. It is assume that the toe oes not slie on the groun, thus the contact can be represente as a revolute joint. This assumption greatly simplifies the ynamics of the stance phase, an it is valiate in section 2.3. To facilitate the comparison between the Linkage Leg uring the stance phase an the SLIP moel, the egrees of freeom of the SLIP are chosen as the generalize coorinates to represent the stance Linkage Leg Lagrangian. As such, the matrices V sw st an G sw st are use to transform the two sets of egrees of freeom, an can be foun in the appenix at the en of the section: 53

63 q LL,sw = V sw st q LL,st Q LL,sw = G sw st Q LL,st (3.15) Using these transformation matrices the new Lagrangian for the Linkage Leg uring the stance phase is obtaine: L LL,st = ( δ(q T LL,st) ) (V δ(t LL,st ) sw st ) T V LL,sw (V sw st ) ( δ(q LL,st) ) + δ(t LL,st ) (3.16) G LL,sw G sw st Q st + L K K LL where K LL is the spring constant of the Linkage Leg, an L K is the change in the length of the segment O 1 O 5 (seen in Fig. 5), which is a close approximation of the spring length. The next step in the Dimensional Analysis stage is to obtain the Pi-factors from the imensionless Lagrangians. This step is shown in [71] to compare ouble penulums inepenently of size, an in [26] to compare the Linkage Leg an the ouble penulum. The Pifactors of the ouble penulum system, the Pi-factors of the SLIP moel an the Pi-factors of the Linkage Leg for each phase are shown in the appenix at the en of the section. The Pi-factors of the AS system mechanisms are inepenent of the state space vector (i.e. configuration of the robot) but the Pi-factors of the Linkage Leg are, however, epenent on the state vector, so they change for ifferent knee angles θ 2. As such, they are average for the range of θ 2 angles expecte uring the swing phase. In orer to achieve ynamic similarity between the Linkage Leg an the AS, the respective Pi-factors must be equal. In orer to achieve physical analogy between the linkage leg an ouble penulum, there shoul be a one-to-one corresponence between their Pi-factors. Therefore, those Pi-factors of the linkage leg that o not show in the imensionless Lagrangian for the ouble penulum must be zero, an the remaining Pi-factors must be equal to their ouble penulum counterparts. Consequently, the following equalities are hel for the Pi-factors: M 4 = 0, M 5 = 0, M 6 = 0, M 7 = 0, M 8 = 0, M 9 = 0, M 14 = 0, M 15 = 0 (3.17) M M = N M, M 1 = N 1, M 2 = M 10 = N 2, M 3 = M 11 = N 3, M 12 = N 4, M 13 = N 5 (3.18) Note that if the linkage leg is parallel, the Pi-factors are constant, an all the equalities in Equation 3:17 are satisfie. Therefore, a parallel linkage leg that satisfies the equalities in Eq. (19) can behave exactly as a chosen ouble penulum. For a generic linkage leg, however, the Pi-factors are not constant, but they vary with θ 2. It was seen in section 3.1 that the Linkage Leg uring stance is similar to a SLIP moel. 54

64 The SLIP moel optimize in section 3.2 has the prismatic joint aligne with the virtual leg (i.e. the line connecting the toe with the hip joint). A parallel Linkage Leg cannot approximate a SLIP moel, because the irection of motion of the toe with respect to the hip, shown by the arrow in figure 3.19, is not aligne with the virtual leg. A Linkage Leg where the intersection of O 1 O 2 an O 4 O 5 is coincient with the groun plane when the virtual leg is vertical will have the same kinematics as a SLIP moel. If the leg angle chances the alignment will be only approximate. As such, an optimization proceure is use to achieve the best balance between satisfying equations (3.17), (3.18) to achieve a close approximation to the ouble penulum, an satisfying the ynamic an kinematic constraints impose by the SLIP moel. The parameters to optimize are shown in Table A.1 in the Appenix, together with the range use an with the final optimize values. The geometry an the ynamic parameters of the leg is the same as the one use in section 3.1 bon graph stage. The lengths a 5 an a 6 are not use in the optimization, but they are instea calculate separately to ensure the point G in Figure 3.19 is on the groun level for the nominal leg length. A multigoal optimization using the fgoalattain algorithm in Matlab is use to achieve the optimal linkage leg shown in Table A.1. The goal was a vector forme by the equations (3.17), (3.18) an the ifference between the respective Pi-factors of the SLIP moel. Figure 3.19 Parallel Linkage Leg (left) an SLIP-like Linkage Leg (right) 55

65 Figure 3.20 Gait of the optimally analogous Linkage Leg Figure 3.21 Comparison between the behaviours of the optimal AS system an the ES (LL) optimally analogous to it. The gait of the Linkage Leg optimally analogous to the AS obtaine in section 3.2 is shown in Fig 3:20, an is compare with the gait of the optimize AS in Fig The esire trajectories an PD controllers for the controlle egrees of freeom (θ L, θ 2, an θ B ) are the same for the AS an the Linkage Leg. As it can be seen in Fig. 3:21, the behaviour of the two systems is reasonably 56

66 close, inicating that a goo analogy is achieve by the imensional analysis stage, an the optimal joint trajectories obtaine for the AS can be irectly applie to the Linkage Leg. The ES is able to achieve a repeatable gait aroun the same Poincare point, optimal trajectories an low- an top-level controllers. The similarity in the ynamic behaviour of the Linkage Leg to that of the AS means that the control system evelope base on the AS can be applie to a Linkage Leg without changes. This is an important result, as it inicates that the mechanical implementation of the leg can be ifferent than the moel use in the control system. The ynamic moel can be optimize to make the control of the running gait simple an intuitive, an then the mechanical implementation of the leg can be optimize for simple construction, light weight, or other such criteria. Furthermore, the analogy between the mechanical implementation an control moel can hol for multiple ifferent moels. This means that each phase of motion can have a ifferent optimal moel which follows a ifferent behaviour, an the Dimensional Optimization proceure outline is able to fin a single mechanical system that achieves a behaviour optimally similar to each of these mechanisms. 3.5 MbA Linkage Leg conclusions The goal outline by the MbA methoology is achieve for the case stuy of the Linkage Leg. The Bon Graph stage was able to fin much simpler mechanisms that have very similar behaviour to the Linkage Leg for each of the motion phases. The SLIP moel is a goo approximation uring the stance phase, an the ouble penulum is a goo approximation uring the swing phase. This simplification was achievable because the behaviour was consiere only uring the specific motion phase, so generality was not a require property of the analogy. The mechanical parameters an the control strategy are simultaneously optimize in the subsequent Analogous System Optimization stage. The system is simulate as if the robot leg consists of a SLIP moel uring stance an of a ouble penulum uring the swing phase. Two transition functions switch the system between the two motion phases, making the touchown an takeoff transitions simple to simulate an easy to unerstan intuitively, an are valiate against a more realistic Simulink moel with groun contact in chapter 2. The Analogous System is able to achieve a spee of 5m/s using less than 50Nm peak torque an showe a stable repeatable gait. The goal of the thir stage, the Dimensional Optimization stage is to achieve a mechanical esign of the Linkage Leg such that it behaves like a SLIP moel while on the groun as similar to a ouble penulum while in the air. To achieve this goal the theory of ynamic similarity is 57

67 use. The Lagrangians of the SLIP moel, ouble penulum an Linkage Leg are expresse in imensionless forms, an Pi-factors are calculate. If the corresponing Pi-factors of the Linkage Leg an each of the mechanisms forming the Analogous System are equal, then the Lagrangians will be equal inepenent of the system state, an as such will result in similar equations of motion. The requirements impose by the ouble penulum an the SLIP analogies are not able to be simultaneously exactly achieve. An optimization proceure is use to fin the Linkage Leg that results in the closest behaviour to both the simpler mechanisms, an the resulting Linkage Leg gait is similar to that of the Analogous System when the control strategy an controller gains obtaine base on the AS are applie to both systems. In the following chapter the Linkage Leg obtaine using the MbA methoology is compare with the Linkage Leg obtaine by a competing strategy, Reuction-by-Feeback. 58

68 4 Comparison between Mechatronics-by-Analogy an Reuctionby-Feeback Mechatronics by Analogy is able to reuce the esign space for a legge robot, an achieve a goo performance when applie to the case stuy of the Linkage Leg presente in the previous section. Legge robots are a goo test case for mechatronic esign methoologies, as they require a close interaction between the mechanical esign an the control strategy [21]. A lot of work has been one to fin simple moels that are able to represent the behaviour of walking machines, either robots, humans or animals [21], [27]. As such, methos that simplify the ynamic behaviour of real mechanisms are very useful in the esign of legge robots. Mechatronics by Analogy was alreay shown in Chapter 3 to achieve this goal, an in this section it will be compare with a competing methoology, which uses a control layer to force the leg to have the esire ynamic behaviour. A top level control layer then controls the gait of the robot assuming it has the ynamics of the simpler system. This methoology is referre to as Reuction-by-Feeback (RbF) here an by its authors [18]. 4.1 General escription of Reuction-by-Feeback methoology Reuction-by-feeback is a control methoology use that creates an embee simpler mechanism which achieves the esire gait. This is achieve using kinematic couplings between the joints in the robot (epenent egrees of freeom) an the unactuate egrees of freeom (inepenent egrees of freeom). This effectively creates a esirable mechanism whose hybri zero-ynamics have a esirable gait pattern an achieve passive running. Of course, actuator inputs are necessary to achieve the kinematic couplings, so the system is still close loop. Once the mechanism with esirable hybri zero ynamics is esigne, the kinematic couplings between the egrees of freeom can be finely tune to eal with outsie isturbances, but the nature of the kinematic couplings shoul allow for better isturbance rejection than traitionally controlle joints [18]. There are two control loops to the reuction-by-feeback methoology: the inner-loop control, which achieves the esire kinematic couplings between the epenent egrees of freeom an the inepenent egrees of freeom, an the outer loop control, which changes the couplings epening on external isturbances. The main benefits of the reuction-by-feeback approach is the ability to construct the kinematics of the zero ynamics mechanism any way the esigner esires. This allows for an 59

69 intuitive unerstaning of the mechanism behaviour, an it allows the esigner to buil up the theoretical mechanism in stages. Furthermore, if an inepenent egree of freeom suffers an unexpecte change of position or velocity ue to an outsie isturbance, the system reacts instantaneously because of the kinematic couplings between that inepenent egree of freeom an the epenent egrees of freeom of the robot. The main issues with the reuction-by-feeback approach is that the esigner has no way of efining the ynamics of the system. The mechanism is efine by the kinematic couplings between joints, an the esigner cannot influence the ynamics irectly, in an intuitive fashion. 4.2 Reuction-by-Feeback applie to the Linkage Leg The Reuction-by-Feeback (RbF) metho is applie to the Linkage Leg presente above, following the same steps as presente in [18], where it was applie the control of Thumper, a single legge running robot. The control in [18] is base on the iea of embeing a simple mechanism by esignating virtual mechanical constraints between the controlle egrees of freeom an the uncontrolle egrees of freeom, an thus reuce the behaviour of Thumper to that of a simpler mechanism. The RbF Linkage Leg is simulate just like the Linkage Leg use to test the Mechatronics by Analogy (MbA) methoology, presente in chapter 3 an in Figure 3.1. The superscript RbF was ae to the parameters of the RbF Linkage Leg to ifferentiate them from the ones use previously. For the MbA Linkage Leg, the swing an stance phases are simulate inepenently, using the f sw an f st functions, shown in Chapter 2. The inputs to these functions are the current state of the robot an the joint torques, an the output is the erivative of the robot state. For the RbF methoology, two virtual constraints are impose to the positions of the controlle egrees of freeom. During the swing phase 1,sw = 1 RbF (x 0 RbF ), 2,sw = 2 RbF (x 0 RbF ) where x 0 RbF is the istance travele by the hip from the previous takeoff position, as seen in figure 4.1. For the stance phase B = B RbF ( L RbF ), where L RbF is the angle of virtual leg (line connecting the toe point an the hip joint) of the Linkage Leg, show in figure 4.1. The x 0 RbF an L RbF inepenent variables were chosen similarly to [18], as the robot presente there is also a single legge running robot. The 1 RbF an B RbF functions are chosen to be fourth orer Bezier function: RbF 1 (x RbF 3 3! 0 ) = (P 1 RbF) k k=0 ( x RbF 0 k!(3 k)! x 0,sw,f RbF ) k (1 ( x 0 RbF 3 k x RbF )) 0,sw,f 60

70 RbF B ( RbF 3! L ) = (P B RbF) RbF ) (1 ( L RbF RbF 3 L,st,i k=0 RbF )) (4.1) k ( RbF RbF L L,st,i k!(3 k)! RbF L,st,f L,st,i k RbF L,st,f L,st,i RbF where x 0,sw,f is the expecte istance travele by the hip uring the swing phase, an it is base 3 k on the estimate of the time spent in the air an the spee at takeoff. For the stance phase RbF L,st,i RbF x 0,sw,f RbF = x 0,sw,i RbF (2 y 0,sw,i g) estimate swing time is the virtual leg angle at touchown, an RbF L,st,f the virtual leg takeoff angle, base on the stiffness of the virtual leg, K vr. is the estimate of RbF L,st,f RbF = L,st,i K vr m B +m 1 + (m 2+m4 ) 2 (π ) estimate stance time The vectors P 1 RbF an P B RbF are the Bezier parameters vectors [85]. For a fourth egree Bezier equation the initial position, initial velocity, final position an final velocity can be specifie inepenently. As such, P RbF 1 = [θ 1,sw,i θ 1,sw,i + θ 1,sw,i θ 1,sw,f + θ 1,sw,f θ 1,sw,f ] an P RbF 1 = [θ B,st,i θ B,st,i + θ B,st,i θ B,st,f + θ B,st,f θ B,st,f ], where the superscript inicates the esire values at the en of the respective phases of motion. The knee constraint uring the swing phase is a fifth orer Bezier function, with the parameters P RbF 2 = [θ 2,sw,i θ 2,sw,i + θ 2,sw,i θ 2,sw,inter θ 2,sw,f + θ 2,sw,i θ 2,sw,i ]: RbF 2 (x RbF RbF k 0 ) = 4 4! (P 2 RbF) RbF ) (1 ( x 0 RbF k=0 RbF )) (4.2) k The extra parameter θ 2,sw,inter ( x 0 k!(4 k)! x 0,sw,f x 0,sw,f 4 k is necessary to give the knee the possibility to retract uring swing, an to prevent the toe from impacting the groun. Note that the final position of the knee at the en of the swing is the same as the initial position at the beginning of the swing. The virtual constraints 1 RbF (x 0 RbF ), 2 RbF (x 0 RbF ) an B RbF ( L RbF ) are thus efine by the esire values at the en of the respective phases of motion, an by the knee intermeiate position, gathere in the vector α RbF : α RbF = [θ 1,sw,f θ 1,sw,f θ 2,sw,i θ 2,sw,inter θ B,st,f θ B,st,f τ θ2 ] The last parameter in α RbF is the constant torque applie to the knee joint uring the extension part of the stance phase, similar to the knee torque applie in Chapter 3. The construction of the 61

71 Linkage Leg, escribe in Chapter 2, is not able to control the position of the knee uring the stance phase, but only apply a constant torque. Given the virtual constraints the swing an stance functions for the RbF Linkage Leg become x sw RbF = f RbF sw (x RbF 0, α RbF, x sw,i ) x st RbF = f RbF st ( RbF L, α RbF, x st,i ) where x sw,i is the initial state of the system of the swing phase, an x st,i is the initial state for the stance system. The transformations st sw an sw st between the swing an stance phases are similar to the ones use in the MbA Linkage Leg simulation an are shown in below at the en of this chapter. RbF states that an embee mechanism, efine by the holonomic constraint equations (4.1) an (4.2) is simpler than the original Linkage Leg an it is efine by the parameters α RbF [18]. These parameters nee to be chosen such that the embee mechanism results in a gait that is esirable. To obtain a goo comparison with the Linkage Leg obtaine in chapter 3 the same optimization proceure is use. The constraints an cost function are show in Table 3.6. The optimization variables are shown in Table 4.1 at the en of this section, with the initial an final optimize values. The initial Linkage Leg parameters are the same ones from Table 3.3, which were use to obtain the bon graphs shown in section This ensures that the comparison between the MbA an the RbF is correct, as both methos start from the same initial Linkage Leg. Once the constraint parameters α RbF are optimize the secon step of the RbF methoology esigns an outer control system that tunes the parameters α RbF base on the error in each step. This is necessary because the holonomic constraints that form the embee mechanism are able to achieve the esire gait for a specific x sw,i starting configuration. If the robot eviates from the specific starting configuration the embee mechanism will not be able to achieve the esire step unless the embee mechanism is moifie to account for it. The metho chosen in [18], as well as in section 3.2.1, is to esign a Linear Quaratic Regulator (LQR) which changes the parameters α RbF epenent on the initial state of the system at the RbF beginning of each step. The proceure use to etermine the LQR gain K LQR for the RbF Linkage Leg is very similar to the one previously escribe for the Analogous System in 3.2. It involves calculating the Jacobian of the system equation with respect the control parameter vector α RbF an to the initial state vector x sw,i. 62

72 x st,i = sw st (x sw,f ) Swing Phase x sw RbF = f RbF sw (x RbF 0, α RbF, x sw,i ) Stance Phase x st RbF = f RbF st ( RbF L, α RbF, x st,i ) x sw,i = st sw (x sw,i ) x sw,i Figure 4.1 Reuction-by-Feeback Linkage Leg The resulting gait of the RbF Linkage Leg is show for three steps in Figure 4.2. The gait obtaine is similar to the one obtaine using the RbF metho in [18]. Figure 4.3 shows the behaviour of the Linkage Leg starting from the first step. The first three rows show the horizontal position of the hip joint, vertical position of the hip joint, boy angle with respect to the groun in 63

73 the left column, an their erivatives in the right sie column. The bottom row shows the torques of the hip (left) an the knee joint (right). It can be seen that the top level control ajusts the α RbF constraint parameters to account for the errors in the initial optimization, likely ue to the constraints in table 3.6 not being perfectly satisfie. Figure 4.4 shows the virtual kinematic constraints (Equations 4.1 an 4.2) for the gait obtaine using RbF methoology. As it can be seen, the hip an knee torques are able to keep the virtual constraints satisfie. The constraint B RbF ( L RbF ) has to be ajuste in the range of L RbF [82 0, 85 0 ] because this is beyon the expecte maximum virtual leg angle RbF L,st,f, likely ue to the stance time being longer than the estimate. The gait velocity is limite by the maximum torque uring the stance phase. A faster gait generally results in a shorter stance phase, as shown in [20] an [53], which means that the stance phase virtual constraint experiences a higher virtual reaction torque (that must be supplie by the hip actuator). The constraints in table 3.6 come into effect for torques higher than 50Nm meaning the optimization must reuce the top spee to keep the torque from becoming too high. This is especially significant for the hip torque, which must rotate the robot boy, the part with the highest moment of inertia) back to its takeoff position. Furthermore, the optimize RbF Linkage Leg oes not exhibit the uncouple property of the MbA Linkage Leg, explaine in Chapter 3. This means that the force exerte by the leg on the boy is not aligne with the hip toe virtual leg, an thus creates a torque aroun the hip. This torque must be counteracte by the hip actuator torque. Higher spees generate higher forces in the leg [53], hence higher torque that must be counteracte, an Figure 4.2 Reuction-by-Feeback Linkage Leg gait 64

74 the optimization algorithm use (fmincon in Matlab) was not able to fin a ecouple leg architecture. Figure 4.3 Reuction-by-Feeback Linkage Leg hip trajectory an joint torques Figure 4.4 Reuction-by-Feeback Linkage Leg virtual joint constraints 65

75 4.3 Comparison between MbA Linkage Leg an RbF Linkage Leg The Linkage Leg obtaine using the RbF methoology is compare with the Linkage Leg obtaine using the MbA methoology for three steay state steps, an it is shown in Figure 4.5. The three steps of the ifferent legs have ifferent urations, an have been alighte to start at the zero time mark. As it can be seen, MbA results in a higher spee an a lower torque, both uring the swing an the stance phases. The optimization metho use to obtain the RbF an MbA Linkage Legs was able to obtain a faster gait for the simpler Analogous System use in the MbA than for the more complex Linkage Leg mechanism an the RbF embee mechanism. Specifically, the higher hip torque require by the uncouple RbF linkage leg limits its achievable top spee. The Knee torque is also much higher for the RbF Linkage Leg, inicating that the useful retraction mechanisms presente by ouble penulum ynamics are not present to help keep the torque low. The much lower torque requirements uring stance phase of the MbA Linkage Leg are explaine by the presence of the user-efine constraint that the Linkage Leg is ecouple, that is, Figure 4.5 MbA Linkage Leg compare with RbF Linkage Leg 66

76 the force ue to the leg spring oes not generate a torque at the hip, as explaine in.figure The ecoupling property has shown beneficial properties for other SLIP-like robots [20]. The RbF optimization oes not have a mechanism to insert such constraints, an is not able to fin a ecouple configuration. This results in the hip joint requiring a high torque to counteract the moment ue to the force of the leg spring. In conclusion, the ability of the MbA methoology to extract meaningful an transparent mechanisms from the Linkage Leg concept, an then give the esigner the ability to refine the esign base on prior knowlege about these simpler mechanisms resulte in a lower torque requirement an thus higher possible spees Disturbance rejection comparison between MbA an RbF Linkage Legs One of the main avantages of the RbF methoology is that its embee mechanism relates the inepenent egrees of freeom to the controlle egrees of freeom using virtual constraints. This causes an instantaneous change in the velocity of the controlle egree of freeom (to the limits of the actuator torque) when the inepenent egree of freeom experiences an instantaneous change in velocity, such as ue to an external isturbance. The trajectories chosen for the MbA methoology are time epenent, an as such will follow the pre-programme esire trajectory inepenent of the external isturbances. Once the step is complete, the trajectories are recalculate base on the isturbances encountere, as explaine in section However, some isturbances might be too great to allow for the step to be complete, an the gait will fail. The MbA an RbF Linkage Legs are compare below for a terrain isturbance that simulates a step 0.1m on the groun. The isturbance happens at touchown, so both robots must eal with the stance phase before they have a chance to ajust their gaits at takeoff. The gaits of the RbF an MbA Linkage Legs show that the robots take a few steps to recover from the isturbance. The RbF robot shows less eviation from the original gait pattern, especially for the angle of the boy B an height of the hip joint y 0, but both robots take aroun 5 steps to return close to their original gait. This inicates that the RbF is to some extent better than the MbA at ealing with isturbances, as expecte. In conclusion, the Reuction-by-Feeback approach was applie to the esign of a Linkage Leg similar to the one obtaine using the MbA methoology. The RbF Linkage Leg isplaye slightly less eviation from its optimal gait than the MbA Linkage Leg if isturbe, but at the cost of a greatly reuce performance. The MbA Linkage Leg was able to achieve 5m/s using less than 67

77 50Nm of maximum actuator torque, whereas the RbF Linkage Leg achieve 2.3m/s using a maximum of 150Nm of torque. 0.1m step. MbA Linkage Leg Moment of 0.1m step. RbF Linkage Leg Figure 4.6 MbA Linkage Leg compare with RbF Linkage Leg for a 0.1m step isturbance 68