On the Control of Gait Transitions in Quadrupedal Running

Transcription

1 On the Control o Gait Transitions in Quadrupedal Running Qu Cao, Anthom T. van Rijn and Ioannis Poulakakis Abstract This paper examines the problem o transitioning between gaits using a gittal-plane reduced-order model o quadrupedal running, which has compliant elements in its torso and legs. First, periodic motions that correond to pronking and two variations o bounding are generated. Analysis o the torso pitch motion in these gaits reveals basic dierences that determine whether transitions between these gaits can be achieved. This observation is then incorporated in control design to enhance the stability o the generated motions. The domain o attraction o the closed-loop systems is estimated through simulation, showing that transition between pronking and bounding can be realized as a sequence o switchings between ixed points. The results in this work can be regarded as the irst step towards synthesizing controllers or gait transitions within template models or quadrupedal running gaits. I. INTRODUCTION Quadrupedal animals exhibit various gait patterns, including pronking, trotting, bounding and galloping. Evidence rom research in biomechanics suggests that each gait pattern is used over a limited range o eeds and that switching between dierent gaits occurs at eciic eeds [1] to reduce the metabolic cost [1] or the bone stress [2]. Early studies classiied quadrupedal gaits with reect to the duration each leg ends on the ground using a ootall ormula [3] or a gait diagram [4]. These intuitive observations were later utilized in the development o quadrupedal robots that can switch rom trotting to pacing by regulating leg touchdown during light at low eeds [5]. Subsequent research in robotic legged locomotion concentrated almost exclusively on realizing gait transition using neurally inired controllers [6] [9], with central pattern generators (CPGs) being a common theme. CPGs are neural networks that produce coordinated patterns o rhythmic activity; or example, dierent inter-limb coordination patterns can be realized with simple input signals and without rhythmic eedback as explained in [10]. By tuning the phase dierence between the legs, transitions between walking and trotting have been realized in several quadrupedal robots [6] [9]. However, the generated motions are at low eeds and not highly dynamic. This work is supported in part by NSF grant CMMI and ARL contract W911NF , and by the European Union (European Lielong Learning o the National Strategic Reerence Framework (NSRF)) ResearchFundingProgram:ARISTEIA:Reinorcementotheinterdisciplinary and/ or inter-institutional research and innovation. Q. Cao and I. Poulakakis are with the Department o Mechanical Engineering, University o Delaware, Newark, DE 19716, USA; {caoqu, poulakas}@udel.edu. A. T. van Rijn is with the Department o Mechanical Engineering, Eindhoven University o Technology, Eindhoven, The Netherlands; a.t.v.rijn@student.tue.nl. S 0 D 0 Γ 1 z 0 z 1 S 1 D 1 φ 0 φ 1 Fig. 1. A conceptual illustration o the transition between two dierent limit cycles, i.e., φ 0 and φ 1,whichinleggedlocomotioncouldcorreond to periodic motions o dierent gaits, or instance, pronking and bounding in this study. S 0 and S 1 are the Poincaré sections o φ 0 and φ 1,and z 0 and z 1 are the correonding ixed points. D 0 and D 1 are the domain o attractions at the Poincaré sections or φ 0 and φ 1,reectively. In this paper, we ocus on investigating gait transitions in quadrupedal running using a reduced-order model a template in the terminology used in [11] which can capture the dominant eatures o dierent gaits without delving into the ine structural characteristics o a robot (or an animal). Aseriesotemplatemodelsodierentmorphologieshave been proposed to study symmetric quadrupedal running gaits in a reductive setting [12] [16]. However, the majority o these models study gaits individually; amodelcanperorm one or several types o gaits, but no transitions between these gaits are examined. In this work, however, we turn our attention to the easibility o gait transitions by analyzing the dynamic and the stability properties o dierent dynamic quadrupedal gaits. Our objective is to provide eedback controllers that guarantee stable switching rom a source to atargetedgaitpattern. In more detail, gait transition is ormulated as a problem o switching between limit cycles as is conceptually illustrated in Fig. 1. The limit cycles, φ 0 and φ 1,representperiodic motions correonding to dierent gaits; in this study, we restrict our attention to pronking and bounding. To characterize stability, the method o Poincaré is used, resulting in two discrete-time mappings P 0 and P 1 and the correonding ixed points z 0 and z 1.Assumingthetwolimitcyclere stabilized under the inluence o the controllers Γ 0 and Γ 1, reectively, the domain o attraction o the limit cycles on the Poincaré sections S 0 and S 1 can be estimated as D 0 and D 1.Byexaminingtherelationshipbetweentheestimated domains o attraction and the ixed points, easible transitions can be determined. For instance, as shown in Fig. 1, i z 0 is within the domain o attraction o φ 1 at the Poincaré section, then by employing the controller Γ 1,thestatesstartingrom

2 z 0 will eventually converge to the orbit φ 1,i.e.,switchto z 1. Γ Mathematically, i z 0 D 1,then z 1 0 z1.furthermore,i z 1 D 0,thentwo-waytransitionscanberealizedbetween Γ the two gaits, i.e., z 1 0 Γ0 z 1. II. MODEL AND GAITS A. Sagittal-plane quadrupedal model Since both pronking and bounding are symmetric gaits, gait transition can be studied using a reductive model in the gittal plane; see Fig. 2. The torso o the model consists o two identical rigid bodies; one represents its posterior and the other its anterior part. The two rigid bodies are connected through a rotational ring to introduce lexibility in the torso. The anterior and the posterior legs are assumed to be massless rings and their contact with the ground is modeled as an unactuated, rictionless pin joint. The mechanical parameters o the model can be ound in [15] and are omitted here. B. Gait description In the pronking gait, shown in Fig. 3(a), both the anterior and the posterior legs touch and leave the ground in unison. In the bounding gait, shown in Fig. 3(b), two variations in the ootall pattern are considered. In the irst variation, which is reerred as bounding with double stance, theposteriorleg touchdown occurs directly ater the anterior leg touchdown, thus there is a part o the cycle where both legs are in stance. In the second variation, which is reerred as bounding without double stance, theposteriorlegtouchdownhappens ater the anterior leg lito, thus the posterior and anterior stance phases are separated by a double light phase. For both pronking and bounding, depending on the state o the legs stanceorlight wedistinguishtheollowingphases:the double light phase, denoted by, in which both legs are in the air; the stance-posterior phase, denoted by, when only the posterior leg is on the ground; the stance-anterior phase, denoted by, when only the anterior leg is on the ground; and the double stance phase, denoted by, in which both legs are in contact with the ground. C. Dynamics in continuous time For the anterior and the posterior stance phases, i.e., i {, }, theconigurationaceq i can be parameterized Fig. 2. L ϕ p γ p (m, J) θ p (x p,y p) l p,k leg k torso θ a l a,k leg ϕ a γ a A gittal-plane quadrupedal model with a lexible segmented torso. Double Flight () Posterior Stance () Posterior Touchdown Double Flight () Fig. 3. Double Touchdown Double Lit-o (a) Posterior Lit-o Anterior Lit-o Double Stance () Posterior Touchdown Anterior Lit-o (b) Double Stance () Double Flight () Anterior Touchdown Anterior Stance () (a) Pronking; (b) Two variations o bounding. by the length l a,l p R o the leg in contact with the ground and its relative angle ϕ a,ϕ p S 1 with reect to the torso, together with the pitch angles (θ p,θ a ) S 2 o the posterior and the anterior parts o the torso, reectively.the coniguration ace Q i o the double stance phase, i.e., when i =,isselectedasinthestance-posteriorphase.forthe light phase, i.e., when i =,theconigurationaceq i can be parameterized by the Cartesian coordinates (x p,y p ) R 2 o the center o mass o the posterior part o the torso and the pitch angles (θ p,θ a ) S 2 o the two segments o the torso. In summary, (l p,ϕ p,θ p,θ a ) Q i or i {, }, q i := (l a,ϕ a,θ p,θ a ) Q i or i =, (1) (x p,y p,θ p,θ a ) Q i or i =. Through the method o Lagrange, the dynamics o the model in each phase can be written in state-ace orm as ẋ i = i (x i ), (2) evolving in TQ i := {x i := (q i, q i ) q i Q i, q i R 4 }. Note that at this stage, no actuation is employed in the system; the system is passive and energy conservative. D. Event-based transitions Phase-to-phase transitions occur according to Fig. 3, and are triggered by leg touchdown and lito events. The light phase terminates when the vertical distance between the toe o either the posterior or the anterior leg and the ground becomes zero. To realize this condition, the light state vector x is augmented with the parameter array α =(γp td,γtd a ) A containing the absolute touchdown angles o the posterior and the anterior legs; see Fig. 2. In addition, due to the assumption o negligible leg mass, transition rom stance to light occurs when the leg ring obtains its natural length.

3 E. Hybrid Dynamics The dynamics o bounding and pronking can be described by concatenating the continuous-time phases according to the sequence in Fig. 3. To generate periodic motions, the Poincaré method is used. The Poincaré section is taken at the apex height in the light phase ollowing the anterior stance when the vertical velocity o the inal joint becomes zero; that is, { S apex := (x,α ) X ẏ p + L θ } p cos θ p =0, (3) where X := TQ A.Byprojectingoutthemonotonically increasing quantity x p rom the light states x and substituting ẏ p through the condition deining S apex,the(reduced) Poincaré map P : S apex S apex can be deined as z [k +1]=P (z [k],α [k]), (4) where z := (y p,θ p,θ a, ẋ p, θ p, θ a ).Theproblemogenerating periodic motions then reduces to solving the equation z P(z,α )=0 (5) or physically reasonable values o touchdown angles α. The local stability o the periodic motions can be examined by linearizing the Poincaré return map at a ixed point ( z, ᾱ ) to obtain z [k +1]=A 1 z [k]+b 1 α [k], (6) where A 1 = P/ z z = z,α =ᾱ, B 1 = P/ α z = z,α =ᾱ and z = z z, α = α ᾱ.i the ectral radius o A 1 is less than one, then the periodic motion is stable; note that or the energy-conservative models discussed above, there will always be one eigenvalue located at one. III. PASSIVELY GENERATED MOTIONS Alargenumberopassivelygeneratedixedpointscorreonding to the gaits o Fig. 3 are computed ollowing the procedure outlined in Section II-E. All the ixed points exhibit certain symmetry properties as observed in [15], which will be useul in designing transition controllers. In more detail, or both pronking and bounding, θ p = θ a and θ p = θ a at the apex height. In particular, the pronking motion has zero pitch velocity at this instant, i.e., θp = θ a =0. These symmetry properties acilitate the investigation o the states that distinguish the two gaits at the Poincaré section. Figure 4 shows the apex height, the posterior pitch angle and the posterior pitch rate o the ixed points correonding to the pronking and the two variations o bounding. It can be seen that these three types o motions can be distinguished by the pitch rate o the posterior part o the torso; pronking has zero pitch rate, bounding with double stance has relatively small pitch rate magnitude (< 2.5 rad/s) and bounding without double stance exhibits larger values o pitch rate (in the range [2.5 rad/s, 5rad/s]). The signiicant dierence in the pitch rate shown in Fig. 4(c) clearly demonstrates the natural separation in the dynamics o dierent gaits, and reveals a major challenge in achieving gait transition. In the context o transitioning rom rom pronking to bounding without double stance, the model needs to experience a drastic perturbation in the torso oscillation dynamics, which can easily destabilize the motion. In other words, ollowing the deinition in Section I, i bounding without double stance is regarded as the target and pronking as the starting limit cycle, to guarantee convergence, the domain o attraction o the bounding limit cycle should be large enough to include the states o the pronking motion at the Poincaré section. However, the model in its passive and conservative orm is limited in rejecting disturbances and all the ixed points computed are not stable. As a result, transitioning rom pronking to bounding calls or control laws that stabilize the gaits and ensures that the domain o attraction o the target gait is suiciently large. IV. FEEDBACK CONTROL In this section, a controller that utilizes actuation at the torso joint is developed to stabilize the pronking and bounding motions computed in Section III. A similar controller has been proposed in our previous work [17]. A. An extended gait description To enhance stability, the model is allowed to go through additional phases beyond the nominal phase sequence as it converges to the target motion; see Fig. 5. For example, in the nominal pronking motion, both the anterior and posterior legs touch and leave the ground in unison with the me touchdown angle. However, when the motion is disturbed Height (m) Posterior pitch (rad) Posterior pitch rate (rad/s) (a) 0.6 (b) 6 Fig. 4. The apex height (a), posterior pitch angle (b) and posterioe pitch rate (c) o the ixed points correonding to pronking (blue square), bounding without double stance (green triangle) and bounding with double stance (red circle) at eed [1, 4]m/s. (c)

4 (a) Fig. 5. Extended description o the gaits in the presence o perturbation or pronking (a), bounding with double stance (b) and bounding without double stance (c). The shaded phases are the augmented phases andthe dotted lines represent the possible evolution in presence operturbation. during light e.g., when the anterior pitch angle increases then determining the touchdown angles α that result in simultaneous touchdown and lito requires numerically solving the dynamics o the double stance phase. To avoid such cumbersome computation, the model is allowed to enter either posterior stance or anterior stance beore entering the double stance phase; see Fig. 5(a). Under the control action, the duration o these two augmented phases gradually decreases to zero, resulting in the nominal periodic motion. In Fig. 5(b) and 5(c), the double light and double stance phases are augmented ater the anterior stance or bounding with and without double stance, reectively. B. Hybrid controller The hybrid controller introduces control action on two levels. In continuous time, holonomic constraints are imposed to the system to coordinate according to a passively generated motion the torso s lexion-extension oscillations with the leg s motion during stance. In discrete time, the touchdown angles are updated at apex through a linear quadratic regulator. 1) Continuous-time control: To enable the development o non-conservative corrective orces, the model in Fig. 2 is modiied to incorporate one actuator at the torso joint in parallel with the torso ring. With this modiication, the dynamics in each phase i {,,, } becomes (b) (c) ẋ i = i (x i )+g i (x i )u i (7) where u i is the input torque at the torso joint. In the light phase, the input is zero, i.e., u =0and in the stance phases, i.e., i {,, }, wedeinetheoutput y i := h i (q i ) := (θ p θ a ) h d P,i (q i) (8) where h d P,i (q i) is the desired evolution o the relative pitch angle θ p θ a. Note that in (8), h i (q i ) is a unction o the coniguration variables instead o time, and thereore can be interpreted as a (virtual) holonomic constraint. This constraint is imposed on the system through a PD controller: u i = K P,i y i + K D,i ẏ i, (9) where K P,i and K D,i are selected gains. In (8), the desired evolution h d P,i (q i) is determined by a ith-degree polynomial parameterized by the leg angles that is itted to the passively generated motions as in [17]; i.e., 5 h d P,i (q i)= β i,k s k i (q i), (10) k=0 where β i = {β i,k } k=0,...,5 are the correonding coeicients and s i is deined as s i (q i ) := γmax γ(q i ) γ max γ min (11) where { γp, or i {, }, γ(q i ) := (12) γ a, or i =, and γ min and γ max are the minimum and maximum values o γ in the correonding stance phase o the nominal motion. With the continuous-time control action, the closed loop Poincaré map (4) becomes z [k +1]=P cl 1 (z [k],α [k]), (13) and its linearization takes the orm z [k +1]=A 2 z [k]+b 2 α [k], (14) where A 2 and B 2 are the Jacobian matrices o P1 cl with reect to z and α evaluated at a ixed point, reectively. 2) Discrete-time control: Adiscretelinearquadraticregulator is employed to place the legs during light based on eedback o the states at the Poincaré section to minimize the quadratic cost unction J(z,α )= (z [k] z ) Q(z [k] z )+ k=1 (α [k] ᾱ ) R(α [k] ᾱ ) (15) where z and ᾱ represent the nominal states and touchdown angles, and Q, R are positive deinite matrices. It can be shown that the optimal cost-to-go, J,isgivenby J (z )=(z [k] z ) S(z [k] z ) (16) where S is the ininite horizon solution o the associated discrete-time Riccati equation. The optimal eedback policy is then given by α [k] =ᾱ K(z [k] z ) (17) where K =(B 2SB 2 +R) 1 (B 2SA 2 ) and A 2,B 2 have been deined in (14). 3) Closed-loop system with hybrid control: With the hybrid control action, the closed-loop orm o the Poincaré map (13) can be written as z [k +1] = P cl 1 (z [k], ᾱ K(z [k] z )) = P cl 2 (z [k]) (18) Deining β := {β,β },thentheparametersthatcharacterize a locally stable gait are given in Γ := { z, ᾱ,γ max,γ min,β,k} (19) that contains the inormation o the nominal motion as well as the coeicients and the matrix used in controller design.

5 V. GAIT TRANSITION As an example o gait transitions, this section examines switching between pronking and bounding without double stance. First, the strategy o estimating the domain o attraction o the closed-loop system is introduced. Then, the transition will be illustrated as a sequence o switchings among ixed points with the bounding with double stance as an intermediate gait. A. Estimation o domain o attraction Computing the exact domain o attraction or nonlinear systems is generally intractable, even at low dimensions. An alternative is to estimate the domain o attraction using Lyapunov s method [18]. A unction V (z ) is a valid Lyapunov unction or the discrete-time system (18) i V (z ) is positive deinite and V (z [k+1]) V (z [k]) < 0 in a bounded domain B. Then,asubsetothedomainoattractionoaixedpoint on the (reduced) Poincaré section S apex can ound as B(ρ) := {z S apex 0 V (z ) ρ} (20) where ρ is a positive scalar. Note that the linear optimal cost-to-go unction (16) is already a Lyapunov unction or the nonlinear system (18) since S is positive deinite. Thus by deining V (z ) := J (z ),theproblemoestimatingthe domain o attraction on S apex reduces to max ρ s.t z B(ρ),J (z [k + 1]) J (z [k]) < 0 (21) which can be solved using the sum-o-squares (SOS) method [19]. However, due to the error incurred in approximating the closed-loop Poincaré map P2 cl o (18) by its second-order Taylor expansions, the analytically guaranteed estimate o the domain o attraction or (18) is rather conservative. As an alternative, a simulation-based method is used to provide amorecompleteestimate.giventhetargetixedpoint z,0, other ixed points z,i,i {1, 2, 3,...} are taken as initial conditions and the hybrid controller associated with the target ixed point Γ 0 is applied. I ater 15 strides the error o the states rom the nominal values is less than 5%, then the ixed point z,i is regarded as being in the domain o attraction o z,0, i.e., z,i D 0. Figure 6 shows that or the bounding gait with double stance a large number o ixed point (black) lie inside the domain o attraction o the target ixed point (blue). On the other hand, the sum-osquare algorithm, in its current implementation, only veriies averysmallportion(red).improvingthesosmethodto obtain a more complete characterization o the domain o attraction is currently under investigation. B. Transitions between pronking and bounding Without loss o generality, we explore gait transitions between pronking and bounding at eed 2.4m/s; seefig.7. Note that z,0 and z,1 correond to the pronking motion and the bounding motion without double stance, reectively. It can be easily checked through the simulation-based method described in Section V-A that direct transition cannot be realized in either direction i.e., z,0 D 1 and z,1 D 0 Posterior pitch rate (rad/s) Fig. 6. Fixed points that can be driven to a target ixed point (blue) within the double-stance bounding gait. The black points are testedusing simulation method while the red points are predicted by the SOS algorithm. which conirms the predictions o Fig. 4(c). To bridge this gap, bounding with double stance serves as intermediate gait between the initial and target gaits. With reerence to Fig. 4(c), we irst examine the transition between the two variations o bounding, which exhibits an asymmetry. The bounding gait without double stance can converge to a large number o ixed points correonding to the bounding gait with double stance (enclosed by solid red). On the contrary, only a ew o bounding motions with double stance (enclosed by dotted red) can switch to the bounding gait without double stance and they all have pitch velocity close to the bounding without double stance; see Fig. 4(c). The transition between bounding with double stance and pronking exhibits a similar behavior; the pronking motion can switch to a large range o bounding motions (enclosed by solid blue), but in the opposite direction only a limited number o bounding ixed points (enclosed by dotted blue) are within the domain o attraction o the pronking motion. In addition, the magnitude o the pitch velocity o these bounding motions is approximately zero, which describes the pitch velocity o the pronking motion. Given the aorementioned intermediate transitions, the task o switching between pronking and bounding without double stance reduces to the transition between two ixed points within the bounding gait with double stance; i.e., z,2 and z,3 in Fig. 4(c). Noticing that z,2 D 3 and z,3 D 2,two more intermediate ixed points z,4 and z,5 are introduced to complete the transition path, which is z,0 Γ 2 Γ0 z,2 Γ 5 Γ2 z,5 Γ 4 Γ5 z,4 Γ 3 Γ4 z,3 Γ 1 Γ3 z,1 (22) In Fig. 7, the arrows illustrate the two-way transition between pronking and bounding without double stance. In both directions, the transitions can be completed within 50 strides. It should be mentioned that when only one-way transition is considered, a shortcut route can be selected to decrease the number o strides required. For instance, switching rom bounding without double stance to pronking only needs less Γ than 20 strides via the route z 3 Γ,0 1 z,3 z,1. Finally, we remark that the proposed transition strategy can be used to enlarge the domain o attraction o the ixed

6 Posterior pitch rate (rad/s) z,2 z,5 z,0 z,4 z,3 z, Posterior pitch angle (rad) Fig. 7. The transition between pronking and bounding without double stance in the (θ p, θ p) section. The blue square, red circle and green triangles represent the ixed points correonding to the target pronking, the target bounding without double stance and the bounding with double stance, reectively. The black arrows show one o the transition routes. points, drastically improving the capability o the system to reject large perturbations. For instance, when the bounding motion z,0 is perturbed outside o its original domain o attraction, the states cannot converge back to their nominal values by mere use o the correonding controller Γ 0.Yet, i the states are located within the domain o attraction o z,5,thenfig.7showsthatimplementingγ 5, Γ 2 and Γ 0 sequentially realizes convergence to z,0.inact,thetwoway transition sequence (22) indicates that any states located within the domain o attraction o one o the ixed points in the sequence can be driven to any other ixed point in the sequence by suitably switching the parameters o the controller in (19). This way, the domain o attraction o each ixed point is expanded as the union o the domains o attraction o all the ixed points in the sequence; i.e., i=0,1,...5 D i.thisexpansioncanbeconductediteratively in a ashion similar to the LQR-tree algorithm [20] thereby signiicantly enhancing motion stability. VI. CONCLUSIONS AND FUTURE WORK In this paper, the gait transition between pronking and bounding is studied using a reduced-order quadrupedal model. A large number o cyclic motions are generated passively, showing that the two gaits are distinguished by the rate o torso oscillation at the apex height. To stabilize the passively generated motions, a hybrid controller that coordinates the torso oscillation with reect to the leg movement during stance and updates the touchdown angles at the apex height is developed. The domain o attraction o the closed-loop system is estimated in order to search or possible transitions between dierent ixed points. By using bounding with double stance as an intermediate gait, two-way transitions can be realized between pronking and bounding without double stance. Future work will ocus on two aects. First, the implementation o the sum-o-squares technique in estimating the domain o attraction will be improved so that transitions can be determined in a guaranteed way, without extensively relying on simulations. Second, the transition strategy will be extended to involve more gaits, such as trotting and galloping in order to create more complex running behaviors. REFERENCES [1] D. F. Hoyt and C. R. Taylor, Gait and the energetics o locomotion in horses. Nature, vol. 292, pp , [2] A. A. Biewener and C. R. Taylor, Bone strain: a determinant o gait and eed? Journal o Experimental Biology, vol. 123, no. 1, pp , [3] E. Muybridge, Animals in Motion. New York: Courier Dover Publications, [4] M. Hildebrand, The quadrupedal gaits o vertebrates: The timing o leg movements relates to balance, body shape, agility, eed, and energy expenditure, BioScience, vol. 39, no. 11, pp , [5] M. H. 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