Non-linear robust control for inverted-pendulum 2D walking

Transcription

1 Non-inear robust contro for inverted-penduum 2D waking Matthew Key and Andy Ruina 2 Abstract We present an approach to high-eve contro for bipeda waking exempified with a 2D point-mass inextensibeegs inverted-penduum mode. Baance contro authority here is ony from step position and traiing-eg push-off, both of which are bounded to refect actuator imits. The controer is defined impicity as the soution of an optimization probem. The optimization robusty avoids faing for given bounded disturbances and errors and, given that, minimizes the number of steps to reach a given target speed. The optimization can be computed in advance and stored for interpoated rea-time use onine. The genera form of the resuting optimized controer suggests a few simpe principes for reguating waking speed: ) The robot shoud take bigger steps when speeding up and shoud aso take bigger steps when sowing down 2) push-off is usefu for reguating sma changes in speed, but it is fuy saturated or inactive for arger changes in speed. Whie the numericay optimized mode is simpe, the approach shoud be appicabe to, and we pan to use it for, contro of bipeda robots in 3D with many degrees of freedom. I. INTRODUCTION In the ong run we woud ike to expain, and roboticay reproduce, the efficiency, speed and versatiity of human bipeda ocomotion in various terrains. Here we work towards this goa by designing a controer for waking, focusing on the abiity to avoid faing in eve waking, considering various disturbances. Bipeda ocomotion vioates many common assumptions in basic cassica contro: it is noninear, nonhoonomic [], has discontinuities, and changes governing equations during the motion. Hence there is no generayaccepted cassica-contros approach to stabiizing waking. Here we pursue a hierarchica controer strategy for waking contro. Most baance controers use some sort of hierarchica structure, factoring the contro probem for waking into two parts: a high-eve controer concerned with overa robot position and baance, and a ow-eve controer that deas with the many interna degrees of freedom. The high-eve controer may specify foot pacement, eg ength, and ground reaction forces and torques to achieve baance; whie the ow eve controer determines individua motor commands to reaize these sub-goas. A few exampes of robots that use a hierarchica approach incude Atas [2], Asimo [3], MABEL [4], and the Corne Ranger [5], a of which use a baance controer based on one or another simpe ow-dimensiona mode. For instance, *This work is supported by the Nationa Science Foundation (feow ID: 2638) and the Nationa Robotics Initiative (grant number 3798) Matthew Key is a PhD student in Mechanica Engineering, Corne University, Ithaca NY 485, USA mpk72@corne.edu 2 Andy Ruina is a Professor of Mechanica Engineering, Corne University, Ithaca NY 485, USA ruina@corne.edu stance eg stance foot hip swing eg swing foot Fig.. Point-mass waking mode. A mass is concentrated at a point at the hip. The egs are mass-ess. There are two contros: The reative ange of the egs at hee-strike, and the size of the push-off impuse by the traiing eg just before hee-strike. Uncertainties in sensing, mode, and actuation, as we as actuator imits on push-off impuse and step ength, are discussed in the text. the capture point controer [6] [9] uses the inear inverted penduum (LIP) mode to pan future foot step ocations and a trajectory for the center of mass. These high-eve commands are then reaized by giving commands to individua motors based on inverse kinematics and dynamics of a fu high-dimensiona mode of the robot. Athough these simpe modes are primariy used to make the ocomotion baance probem tractabe, they aso seem to be genuiney good modes for baance contro []. Because of their utiity, however motivated, most successfu simpe modes are based on a point mass at or near the hip, ight or mass-ess egs, and actuation to contro the time and ocation of foot fas and the ground reaction force. The modes tend (sensiby, we think) to negect the effects on baance of upper body motions and the detais of eg swing between steps. Within the cass of point-mass modes just mentioned, there are sti choices. One common choice is to use the inear inverted-penduum (LIP) mode, in which the height of the point mass is hed constant (e.g. with capture point and zero moment point contro). Here, however, we use an inverted-penduum (IP) mode which has constant eg ength instead of constant hip height. The IP mode has not been as frequenty used for waking baance contro as the LIP mode, in part because of the mathematica simpifications of the LIP (e.g. []). We use the IP mode for contro because it uses energy more effectivey than the LIP mode. More precisey, Srinivasan and Ruina [2], [3] used numerica trajectory optimization to find positive-work minimizing gaits on a genera point-mass waker and found that the energy-minima waking gait used constant-ength egs: the IP mode of waking. In this mode the stance eg ength remains constant, and the push-off at the end of the stance phase is (at east approximatey) impusive. This is a powered version of the

2 mid-stance swing down push-off push-off then coision coision swing up mid-stance Fig. 2. One waking step. A step starts when the stance eg passes cockwise through the vertica orientation. The hip then fas to the right. Just before the swing eg hits the ground there is push-off from the ground on the stance eg. Immediatey after push-off, the swing eg hits the ground and becomes the stance eg. The former stance eg eaves the ground and the new stance eg swings up to the start of the next step. In the eftmost and rightmost pictures, the ange of the swing eg is arbitrary (because it is fuy controed). so-caed Simpest Waker [4]. The energy effectiveness of impusive push-off and straight eg waking is aso discussed in Kuo [5] for a simiar mode. Because of the IP s good energetics, it is the basis of our waking robots. In the design of the controer here, however, we do not expicity consider energy, focusing instead on robustness to disturbances and quick return to a nomina gait. Zaytsev [6] has introduced a simpe waking controer for this IP mode, based on finding simpe contro aws that maximize the distance from faiure boundaries. Here, we extend Zaytsev s work by optimizing an arbitrary-form controer that is robust to disturbances and aso stabe. Additionay, the design process here is fast enough that it can be repeated to stabiize a range of speeds thus generating a fu feedback poicy for a range of waking-speed goas. Despite a difference of methods with Zaytsev, there are common resuts (discussed beow). This may indicate that we are extracting features that are necessary aspects of any robust waking baance controer that is reasonaby constrained by actuator and sensor imits. II. WALKING MODEL Our mode consists of a point-mass hip on mass-ess inextensibe egs (Fig. ). The mode has two contro inputs at each step: the ange of the stance eg (φ) at hee-strike, and the push-off impuse (p) that occurs immediatey before hee-strike. A step comprises four distinct phases, starting from a midstance (more or ess a Poincaré Section) where the stance eg is vertica and rotating cockwise. These phases are iustrated in Fig. 2: a swing-down to the step ange (φ); foowed by an impusive push-off (p); then hee-strike and eg-switch; and finay a swing-up to mid-stance. Throughout the paper we measure waking speed in the various phases of the gait by the dimensioness (divide by g/) anguar veocity ω of the stance eg. A. Actuation The step ength is determined by the step ange: step ength = 2 sin φ. We negect swing-eg inertia: its motion does not affect the motion of the stance eg. The step ange is bounded (constrained) to mimic the joint imits of a rea robot. The impusive push-off is meant to be a proxy for the energy injected into the waking motion by the extension of the traiing eg (anke extension). The bounds on the pushoff are a proxy for the maximum power avaiabe for eg extension. B. Equations of Motion The mode is a simpe inverted penduum, but with impuses and resets at each step. The parameters m, g, and represent the robot s mass, gravitationa acceeration, and eg ength respectivey. The contro variabes φ and p represent the step ange and push-off impuse respectivey. The step starts with the stance eg vertica and rotating with speed (ω k ) towards hee-strike. The anguar rotation rate immediatey before push-off can be obtained via conservation of energy: ω = (ω k ) 2 + 2g ( cos φ). () Next, a push-off impuse is appied to the point-mass aong the stance eg, changing the hip s veocity vector. After that, the swing eg becomes the new stance eg as it coides with the ground, exerting another impuse on the point-mass hip. The composition of these two coisions, governed by anguar momentum baance about the new stance foot, yieds the rotationa speed of the new stance eg (ω + ), ω + = (ω ) ( cos 2 φ sin 2 φ ) + 2p cos φ sin φ. (2) m After the two coisions, the stance eg swings up to the next mid-stance, again rued by conservation of energy, ω k+ = C. Feasibiity conditions (ω + ) 2 2g ( cos φ). (3) A waking step is considered successfu (i.e. the robot did not fai) if two conditions are met: first, the stance eg is aways in compression (this is aso a no-fight condition), and second, the speed after the hee-strike must be sufficient to continue on to the next step without faing over backwards. These conditions are expressed with the four inequaity constraints: g ω < cos φ (4) < (2m)(ω ) cos φ sin φ p ( cos 2 φ sin 2 φ ) 2 (5) 2g g ( cos φ) < ω+ < cos φ. (6) The first inequaity (4) is a restriction on the speed before the push-off impuse. The second inequaity (5) is a restriction on the coision impuse - the coision cannot pu the waker towards the ground. The fina inequaities (6) are restrictions on the ower and upper bounds on the speed after coision, preventing both faing over backwards and fight.

3 D. Disturbance Mode In this paper we caim that the controer is robust to errors in modeing (7), actuation (8)(9), and sensing (). Each type of error is modeed as a perturbation: := + δ, δ (7) p := p + δ p, δ p p (8) φ := φ + δ φ, δ φ φ (9) ω k := ω k + δ ω, δ ω ω () We represent the vector of disturbances: δ = [δ, δ p, δ φ, δ ω ], δ D () The set of disturbances D may be thought of as a fourdimensiona hyper-rectange. We define a set of maxima disturbances D max that correspond to the 2 4 corners of this hyper-rectange. D max = {[±, ± p, ± φ, ± ω ]} (2) We use the fu set of disturbances (D) for doing controer verification and testing, and the smaer sub-set of maxima disturbances (D max ) for controer design and optimization. III. CONTROLLER DESIGN Our controer is a function that maps the estimated midstance 2 speed (ˆω k ) to a desired push-off impuse (p) and step ange (φ). When impemented on a rea robot, the push-off impuse is reated to how much energy the extension of the traiing eg adds to the robot before the next step. The step ange corresponds to a target foot-pacement ocation on the ground. {p, φ} = K(ˆω k ) (3) The controer K aims to stabiize the waking gait to a user-specified target speed (ω ). Here, we outine the design process for a singe given target speed. The method is then repeated to find controers for a variety of speeds (see resuts section). We discretize the range of possibe input mid-stance speeds Ω I = [, ω max ]. The output of the controer at each of these grid-points is computed by soving an optimization probem: p and φ are the contros that stabiize the midstance speed (ω ω ) in the fewest number of steps whie preventing fas despite a possibe bounded disturbances. At run-time the controer is evauated via inear interpoation over this grid. We express the design requirements (stabiity and robustness) as constraints in the optimization probem that defines the controer. Thus, if the optimization returns a feasibe soution, then the controer wi satisfy the design requirements precisey for every grid point in the controer. For exampe, [+, + p, φ, + ω] D max 2 Mid-stance is the point during a step when the stance eg is vertica. A. Asymptotic Stabiity We want the controer to bring the robot towards the desired waking speed (ω ), starting from any mid-stance speed ω k drawn from the set of aowed initia speeds Ω I. This goa can be expressed as the need for reduction at each step k of a discrete Lyapunov function (V ), with V defined as the mid-stance speed-error squared: V (ω k ) = (ω ω k ) 2 (4) The controer is asymptoticay stabe if the Lyapunov function decreases at each successive step: V (ω k+ ) < V (ω k ) ω k Ω I (5) This condition (5) is imposed as a constraint in the controer design, thus any controer wi be asymptoticay stabe, in the absence of disturbances. B. Robust Stabiity Asymptotic stabiity in the absence of disturbances is good, but we woud aso ike to show that the controer is sti stabe given any disturbance δ D. In this case, it is not possibe to show convergence to a point, but we can (and do) show convergence to some finite set Ω G that contains the target mid-stance speed. For this aspect of the controer, we separate the controer design and verification. For the controer design we find the contros that minimize the Lyapunov function (4) over the finite the set of maxima disturbances D max. The set D max is a proxy for the disturbance that precisey maximizes the Lyapunov function. Note that ω i is the next mid-stance speed, subject to disturbance δ i. The contros (p, φ) and initia state (ω k ) are hed constant over each step in the sum: f(p, φ) = (ω ω i ) 2 (6) δ i D max Once the controer has been designed, we do stabiity verification by running an additiona optimization to find the precise disturbance that maximizes the Lyapunov function at the next step ω k Ω I. The resut of this optimization is the size of the goa set Ω G that satisfies (7) and (8). V (ω k+ ) < V (ω k ) V (ω k+ ) V (ω k ) ω Ω I Ω g δ D (7) ω k+ Ω G ω k δω G δ D (8) The first of these equations (7) shows that the controer is asymptoticay stabe to Ω G, even in the presence of disturbances. The second equation (8) shows that once the waking speed is inside the boundary of the goa set (δω G ), that there is no disturbance that can push it out.

4 C. Fa Prevention In addition to reaching the target waking speed, we woud ike that any execution of the controer avoid faing. This is accompished by adding the constraints () (3) to the optimization probem, and requiring that they hod for δ D max. D. Impementation The constraints for the optimization probem require soving tota of 7 simuated waking steps: one nomina step (without perturbation), and 2 4 perturbed steps (one for each disturbance in D max ). A of these simuated steps start from the same initia state and use the same contro. For each of these simuated waking steps, the intermediate speeds ω, ω + and next step speed ω k+ are passed as decision variabes in the optimization. This aows the dynamics () (3) and no-faing constraints (4) (6) to be expressed as simpe non-inear functions of the decision variabes. The asymptotic stabiity condition (5) and objective function (6) are both quadratic in the decision variabes. Posing each optimization probem in this way makes it easy to compute gradients and sove quicky using standard noninear constrained optimization packages. IV. RESULTS In this section we present an optima controer, as designed using the framework presented in this paper. A. Design Parameters A parameters in this paper (Tabe I) have vaues that roughy match the Corne Ranger [7]. B. Optimization We used Matab s [8] FMINCON optimization software to sove a optimization probems presented here. The optimization probem that defines the controer is evauated for each point on a grid. The resuts here were computed using a grid of 5 points. Simiar resuts are obtained for grids with fewer points, athough there is a sight degradation in performance due to interpoation errors at runtime. The resuts presented here took approximatey 62 seconds 3 to generate in Matab. Three controers were generated TABLE I PARAMETERS Symbo Name Vaue m mass 9.9 kg g gravity 9.8 m/s 2 eg ength.96 m p max max push-off impuse 2.2 kgm/s φ max max stance ange 3 = 2 rad ω max max mid-stance speed 2.56 rad/s eg ength error bound ±.5 p push-off error bound ±.5 p max φ step ength error bound ±.5 φ max ω mid-stance speed error bound ±.5 ω max (incuding verification), with 5 grid-points each, giving a average time of.4 seconds per grid-point. C. Optimized Push-Off Controer The optimized push-off controer is shown in Fig. 3. It has a reativey simpe form: big push-off to increase speed, and no push-off to sow down. For fast waking, the push-off is saturated at the maximum vaue for much of the domain. no push-off target = sow wak max push-off target = medium wak target = fast wak Fig. 3. Optimized Push-Off Controer. This figure shows the optima push-off controer for three different waking speeds. The genera trend is simpe: too fast no push-off, too sow big push-off. Notice that the actuation is saturated for much of the domain for the fast controer. D. Optimized Step-Length Controer The optimized step-ength controer is shown in Fig. 4. The genera trend is that you shoud take sma steps when you are near the target speed, and bigger steps otherwise. In the absence of push-off (p = ), taking bigger steps increases the energy ost due to coision, and wi sow the waking gait. If the push-off is non-zero, then it is scaed by a term that increases with the step ange. In effect, taking bigger steps aows the push-off impuse to be more effective. This expains the genera trend: if a arge push-off is used, then that term dominates the coision osses, and the waking gait speeds up; if a sma push-off is used, then the coision osses dominate and the waker sows. E. Stabiity and Robustness Given the optima controer shown in Figs. 3 and 4, it is possibe to compute the cosed-oop dynamics of the system, mapping the mid-stance speed from one step to the next. Figure 5 shows this so-caed one-step map for an intermediate speed waking gait. The horizonta axis shows 3 Processor: Inte Core 3.4GHz x 4

5 target = sow target = medium.75 target max disturbance no disturbance target = fast big steps sma steps max disturbance Goa Set Set of a possibe steps guaranteed guaranteed error reduction error reduction Fig. 4. Optimized Step-Length Controer. This figure shows the optima step-ength controer, for three different waking speeds. The genera strategy is to use sma steps for the nomina waking speed, and take arger steps otherwise. The sight corner in the sow- and medium-speed waking controers for high speeds is cause by the no fight constraint (4). the mid-stance speed at step k, and the vertica axis shows the mid-stance speed the next step (k + ). Any given point on the horizonta axis maps to a set of points on the vertica axis, corresponding to the set of reachabe speeds given any aowed disturbance. The purpose of this figure is to visuaize the stabiity of the controer. The diagona dashed ines show the points where the Lyapunov function (4) is unchanged from one step to the next. The entire horizonta axis is the set of initia speeds Ω I, and there is a thin vertica shaded region that shows the goa set Ω G. For initia speeds that are outside of the goa set, the Lyapunov function is aways decreasing from one step to the next. For initia speeds inside the goa set, the Lyapunov function (error squared) might increase, but the next step speed wi never eave the goa set. In the absence of disturbances the controer rejects neary a speed error in a singe step, as shown by the neary horizonta ine abeed no disturbance. In the presence of disturbances the controer is sti quite stabe, reaching the goa set in two or three steps even with the worst possibe disturbances. F. Simuation Test As one fina check, we simuated the cosed oop system for a tota of 6 steps, where the robot was subject to random disturbances, uniformy drawn from D 4. We ran 3 simuations, each consisting of 3 steps and starting from a randomy chosen initia condition uniformy drawn from Ω I. In a cases the mid-stance speed stabiized to the target speed within a few steps, and the mid-stance speed remained Fig. 5. One-step speed map. This figure shows the cosed-oop dynamics for the medium-speed (ω = ω max) waking controer. The horizonta axis gives the mid-stance speed for step k, and the vertica axis gives the mid-stance speed for step k + that is achieved by the robust controer. The horizonta shaded region show a possibe steps that occur. The no disturbance ine shows the behavior of the controer in the absence of disturbances. The boundary of the shaded region, marked with max disturbance, shows maximum possibe speed error at the next step due to a disturbance. The dashed ines show the points where the speed error is unchanged from one step to the next. Notice that for most of the domain of the controer, there is a arge reduction in error from one step to the next, despite disturbances. For exampe, if ω k =.75 ω max, then ω k+ [.4 ω max,.6 ω max]. If there were no disturbances, then ω k+ woud be ω max. The vertica shaded region shows the goa set Ω G that satisfies (7) and (8). within the goa set (Ω G ) on a subsequent steps. That is, despite the disturbances appied at each step, the controer was abe to prevent fas in a cases. V. DISCUSSION We have presented a controer for a simpe mode of waking that is maximay robust, by our measures. This controer can reguate a desired waking speed whie preventing fas due to reasonabe errors in the mode, sensors, and actuation. The controer avoids fas for a reasonabe vaues of disturbed and desired waking speeds. That is, assuming we consider ony forwards waking, the basin of attraction of this controer is maxima with the given noise. The approach here was inspired by robust contro: the controer shoud be robust (never fai) for any bounded disturbance, and the waking speed shoud converge to some goa set. The controer was designed using non-inear 4 The perturbation on eg ength was randomy drawn once at the beginning of each of the 3 simuations, and then hed constant for a 3 steps in each simuation. This was done to more cosey mimic a robot that had a modeing error. The other disturbances were randomy drawn on every singe step.

6 optimization, where these robustness and stabiity requirements were enforced as constraints on the optimization. Athough the resuting controer is correct by construction, we demonstrated the performance of the controer using massive simuation. We designed a controer that coud robusty stabiize sow, moderate, and fast waking gaits for the inverted penduum mode of waking, using parameters based on our robot, the Corne Ranger. An interesting and perhaps genera resut is this: To increase speed, the controer takes arge steps and uses arge push-off. To maintain speed, it takes sma steps and uses intermediate vaues of push-off. To decrease speed, the controer takes arge steps with no push-off. This genera trend is observed across a waking speeds. Athough ony impemented here on a simpe 2D mode, we beieve the approach wi be usefu for a more compex robot in 3D. VI. FUTURE WORK We pan to deveop this controer for use on a 2D robot, Corne Ranger, and then ater, for a 3-D robot with many degrees of freedom. One imitation of our mode is that we restricted it to forward motion without a fight phase. These restrictions make cacuations easier, but are overy conservative with respect to the rea imits on faing; a sma fight phase, or a step backwards does not necessariy ead to a fa. With a more genera mode that aowed some fight and backstepping, we coud make a robust controer with even arger aowed bounds on the various disturbances. The extension from a 2-D robot to a 3-D robot wi raise the mid stance state from number to 2 (or 3 if heading is to be stabiized). The actuation wi go from 2 contros to 3 (push off, step ength and steering ange). However, the addition of more interna degrees of freedom does not change the form of this high-eve controer. Rather the output of a controer of the type deveoped here, wi serve as input to the micro-management of the various joints so as to achieve the desired push off and foot pacement. ACKNOWLEDGMENT This research is supported by the Nationa Science Foundation: Graduate Research Feowship Program (feow ID: 2638) and the Nationa Robotics Initiative (grant number: 3798). Thanks to Petr Zaytzev for his insights about the nature of the contro aws for the simpest waker, and to Anoop Grewa for his work on the simpest waker mode. [4] K. Sreenath, H.-W. Park, I. Pouakakis, and J. W. Grizze, A Compiant Hybrid Zero Dynamics Controer for Stabe, Efficient and Fast Bipeda Waking on MABEL, The Internationa Journa of Robotics Research, vo. 3, no. 9, pp. 7 93, Sept. 2. [Onine]. Avaiabe: [5] P. a. Bhounsue, J. Corte, a. Grewa, B. Hendriksen, J. G. D. Karssen, C. Pau, and a. 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