θ [rad] φ b. [Nm] 20. τ b

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1 Stale Open Loop Walking in Quadruped Roots with Stick Legs M. Buehler, A. Cocosco, K. Yamazaki, R. Battaglia Center for Intelligent Machines, McGill University, Montreal, QC H3A A7, Canada Astract In our previous work we have shown that walking can e implemented on quadrupeds with sti legs and only one actuated degree of freedom in the hip, ased on the concept of controlled momentum transfer. In this paper we show via simulations and experiments on two dierent quadruped roots that dynamically stale and roust walking can e achieved via an open loop controller which is active only during the ack leg support phase. Introduction Most existing four- or eight-legged walking roots are designed for statically stale operation { staility is assured y keeping the machine's center of mass aove the polygon formed y the supporting feet. While this is the safest mode of locomotion it comes at the cost of moility and speed. Furthermore each leg needs at least three degrees of freedom to provide ody support during forward motion. Past research in dynamically stale running roots [] has shown that a dynamic mode of operation permits increased moility, speed, and much simplied mechanical designs y reducing the numer of legs and the actuated degrees of freedom required. We elieve that reducing mechanical complexity, and as a consequence, cost, maintenance, and risk of reakdowns is a critical element towards ringing legged roots into practical applications. For this reason we have started studying and uilding quadrupeds with only one actuator per leg. In [] we have shown that such a design is still capale of a wide range of useful ehaviors, including walking, turning, step/stair climing and running. In this paper, we focus on walking control and show that a very simple open loop leg sweeping algorithm can produce stale and roust root walking. Why is this important? First it is not ovious and of conceptual interest that an inherently unstale dynamic walking motion can e stailized via an open loop algorithm. This might lead to interesting parallels to the iological control of locomotion, where spinal CPGs have een shown to provide \open loop" muscle stimulation without input from the rain [9]. In more practical terms, the presented controller needs no other sensors than foot switches to sense stance, and thus could lead to inexpensive walking roots for toys or entertainment applications. Even in more sensory-rich roots, the availaility of a asic open loop locomotion algorithm might prove critical when sensors fail, or when adverse conditions hamper visual sensing. There exists some pulished research on dynamically stale quadruped walking, which is ased, however, on articulated legs. Miura et al. [8] developed dynamic walking control for their Collie- root. Furusho et al. [5] uilt a quadruped with articulated legs and developed a controller for a running gait. Hirose et al. [6] designed a quadruped walking vehicle for dynamic walking and stair climing. Dunn and Howe [4] presented a smooth walking controller for their iped root, ut the same approach could e used for a quadruped as well. Motivated y McGeer's work [7], Smith and Berkemeier [] showed that quadrupedal walking is possile ased on passive dynamics. Root Model Quadrupeds with articulated legs are capale of a variety of walking gaits, where the swing leg is retracted to prevent toe stuing. When the legs cannot e retracted, like in our simple design, a ounding (rocking) type walking gait produces ody pitching, which provides ground clearance for the swing legs. Thus at any given instant, the root is statically unstale, pivoting either on the front leg or the ack legs. For our model we assume that the exchange etween front and ack legs occurs via instantaneous front or ack leg impacts. Since the front and ack leg pairs move in pairs, a planar model as shown in Fig. suf- ces.

2 at ack leg impact can e calculated as l[?l cos( B + B f ) + L cos B f ] + r? L + ll cos B _ B? +l[?l cos( B + B f ) + L cos B f ] _ B? f = l[l? L cos B ] + r + L? ll cos B or, more concisely, as?l[l? L cos B ] _ B+ = c _ B? + c dot f B? + c _ B+ where the c i are conguration dependent factors. A similar relation can e derived for front leg impacts. 3 Control Figure : SCOUT model The legs are connected to the ody via the ack and front hip joints, H and H f. The two actuators of this planar model are placed at the hips and control the hip angles and f. When a toe, T or T f, is in contact with the ground, it is modeled as a pin joint (no friction or slipping). We assume the leg mass to e negligile, therefore only the supporting leg and its hip actuator inuence the dynamics of the system. Thus, SCOUT can e modeled as an underactuated, inverted doule pendulum with one unactuated joint (stance toe) and one actuated joint (stance hip). The nomenclature is as follows. Suscripts ; f refer to front and ack legs, superscripts B; F denote variales at time of front or ack leg impact, and superscripts +;? denote (velocity) variales just after or efore impact. Thus is the ody angular velocity just after ack leg impact, and B is the ack leg angle at ack leg impact. The equations of motion for each single support phase are derived using the Lagrange Method, and descrie the nonlinear dynamics of the ody angle,, and the hip angle,, as a function of the hip torque,. These equations are used for simulations with Matla, and are provided in [3]. During front or ack leg impact, an instantaneous doule support phase occurs, during which we assume that we can control the hip angles and angular velocities. Based on conservation of angular momentum around the impact toe, the change of ody velocities For purposes of controller design we assume that we can control the hip angles and angular velocities; thus, our inputs are f (t) and (t). However, the simulations (and, of course, the experiments) employ a PD controller to determine the hip torque necessary to track the desired hip angle trajectory. The root states to e controlled are the ody angle and angular velocity _ = : One complete step consists of a front and a ack leg support phase, and a front and a ack leg impact. For purposes of analysis it is convenient to examine the variale of interest,, at only one instant in this step, and we chose the instant immediately following the ack leg impact, n = B+ = B : () Now we can dene a discrete step-to-step return map, S, which maps the ody states just after impact from step n to step n +, as a function of the inputs (t); f (t), n+ = S( n ; (t); f (t)): () For the ramp controller introduced in this paper, the controlled inputs already determine B, so () ecomes a scalar return map for _ B+. The control ojective can e stated as nding continuous hip angle trajectories, (t); f (t), which make the desired ody states,, an asymptotically stale xed point of the return map S.

3 The ramp controller was originally motivated y the desire to nd a controller for dynamically stale walking which requires minimal sensing, computation, and control. Such a controller would keep the front legs xed at f (t) = f, start with a ack leg touchdown angle of B and sweep the ack legs during ack leg stance with a constant angular velocity _ (t) until the front legs impact again. Thus, strictly speaking, our controller is not completely open loop, since we make use of the leg's ground contact sensors to switch the ack leg ramp controller tracking on and o. In addition, the ramp trajectory tracking requires feedack of the ack hip angle and angular velocity. However, it is open loop in the sense that there is no measurement or feedack of the ody angular position (wrt to horizontal) or velocity. These are the states to e stailized, and also the states which are most dicult to measure in practice. F+ θ dot B+ θ dot,n+.5.5 (a).5.5 F φ f [rad].5.5 B+ θ dot,n+ (c).5.5 ().5.5 F φ f [rad] to search for xed points of S in three dimensional controller parameter space. θ [rad] φ [rad] θ dot τ [Nm] time [s] Figure 3: Stale motion produced y the open loop ramp controller. Motor torque limit set to 4 Nm per actuator (8 Nm total for ack leg pair). Simulations (Fig. 3) of the open loop ramp controller suggest that the chosen set point does not only correspond to a xed point of the step-to-step map, ut is at least locally stale. To further investigate the possile staility of this open loop controller around the set point, we added a severe perturation after the rst step, and, to our great surprise, the system converged rapidly ack to the desired set point, = rad=s (Fig. 4) φ,dot Figure : Eect of F f (a) on _ F + and on () _ B+ n+ ; (c) eect of _ on _ B+ n+ A set of MATLAB simulations was used to determine the ramp control design parameters B ; _ ; f which would make our desired ody angular velocity after ack leg impact, = rad=s a xed point of the step-to-step return map, S. To simplify the search, we xed the rst parameter, the ack leg angle at ack leg impact, B = :5rad. In order to otain a large magnitude for the ody angular velocity after front leg impact, the front leg angle should e small (Fig. (a)), ut only a value of f = :3rad will result in the desired = rad=s (Fig. ()). Finally, Fig. (c) shows that there are two ack leg angular velocity settings which achieve the desired = rad=s and we chose the smaller value, _ = :85rad=s. Alternatively, gradient type searches can e employed θ B+ dot step numer Figure 4: System response with open loop ramp controller after severe disturance For a more complete insight into the range of initial ody angular velocities which will converge to the desired set point (the domain of attraction of the controller), we plot the numerical evaluation of the stepto-step return map for the open loop ramp controller

4 in Fig. 5. This plot conrms the unexpected fact that the open loop controller has a domain of attraction which is gloal for all practical purposes, from almost zero initial ody angular velocity, to a maximum ody angular velocity of _ B+ = :3 rad=s, aove which the root would fall over ackwards!.5 lengths and ody length (hip to hip) of :m. Its ody width (left toe to right toe) is :45m at the front and :9m at the ack. SCOUT II [] weighs 7kg and measures :75m in height, :55m in length and :48m in width. SCOUT I has (position controlled) hoy RC servo motors at the joints, while SCOUT II features geared rush DC servo motors. Both roots are controlled y a PC, running the QNX real time operating system []. θ dot,n θ dot,n Figure 5: Numerical evaluation of the step-to-step return map for the open loop ramp controller In order to increase the rate of convergence further, a feedack control mechanism can e added which adjusts the ack leg angular velocity as a function of the initial ody angular velocity. For a range of after-impact ody angular velocities, and a range of ack leg hip angular velocities _, a look-up tale has een generated. The input in this tale is the actual (measured) n and the output is the required _ which, during the ack leg support of the (n + ) step, will result in the desired set point n+ = rad=s. The look-up tale has 8 entries (from n = rad/s to n = :79 rad/s, in steps of. rad/s). For entries dierent from the taulated values, a linear interpolation function is used to generate the outputs. This look-up tale is used to simulate walking for several steps, with 5% error in the (step ) and 3% error for _ F + (step 7). As it can e seen in Fig. 6 the recovery in oth cases is accomplished in just one step. This is conrmed y the closed loop step-to-step return map plot Fig Experiments The open loop ramp controller was implemented on two quadruped roots, SCOUT I (Fig. 8) and SCOUT II (Fig. 9). SCOUT I [] weighs :kg and has leg θ B+ dot step numer Figure 6: Closed loop control achieves one step recovery from disturances θ dot,n θ dot,n Figure 7: Step-to-step return map for closed loop (LUT) ramp controller The controller parameter derivation for the open loop ramp controller as descried aove was done using SCOUT I parameters, ased on a higher angular ody velocity setpoint. The same strong staility properties were otained: Fig. shows transient and steady state experimental walking data - the root converges to the stale xed point operation within two steps, even after large initial errors. More details of the SCOUT I implementations and model verica-

5 3 4 5 6 7 8 9 Step Numer Figure 8: SCOUT I Figure 9: SCOUT II Figure : Stale walking in SCOUT I: Transients from dierent initial conditions; Experimental Data The ramp controller implementation on

5 tion can e found in [3]..5 Angular Velocity (rad/s) Step Numer Figure 8: SCOUT I Figure 9: SCOUT II Figure : Stale walking in SCOUT I: Transients from dierent initial conditions; Experimental Data The ramp controller implementation on SCOUT II uses the control parameter settings derived in the previous section to achieve a stale xed point for the ody angular velocity just after ack leg impact of = rad=s. The conguration at the moments of exchange of support are f = :3rad for the (xed) front leg angle, B = :5rad for the (initial) ack leg angle at ack leg impact, and _ = :85rad=s for the ack leg stance sweep velocity. Experimental data of stale SCOUT II walking is shown in Fig.. The top two graphs show the ody angle oscillations and the angular velocity for six walking steps. The third plot shows the ack leg angle and the tracking error for the constant desired velocity during stance. The ottom plot shows one of the two ack leg actuator torques, limited to 4N m. These experiments conrm the stale and roust walking ehavior expected from the earlier simulations. The plot of angular ody velocities shows a value close to = rad=s at the eginning of the ack leg support phase (marked with 'o'). However, a comparison with the simulation Fig. 3 shows some marked dierences as well. The main dierences are due to our naive assumption of instantaneous exchange of support. Even with SCOUT II's sti legs, there is sucient compliance in the system (toes, transmission elt, gear, motor controller limitations) that we have in fact a doule support phase with accounts for approx. % of the step time! Since we start sweeping the ack legs already during this period, the susequent ack leg sup-

6 θ [rad].4.. θ dot φ [rad] τ [Nm] time [s] Figure : Experimental results for the ramp controller; `*' eginning of the doule stance phase; `o' eginning of the ack leg support and `+' eginning of front leg support; dashed line: desired; solid line: actual port phase is longer. The existence of the doule support phase, and the resulting increase in ack leg support phase explain the large dierence in step times etween simulation (:5s) and experiment (s). The longer step time goes hand in hand with a larger ody oscillation amplitude and peak ody angular velocity. Nevertheless, the qualitative results of the simulations and the experiments are very similar. The fact that we otain stale and roust walking performance despite large modeling errors can e interpreted as another indication of the strong roustness properties of this class of open loop controllers for dynamically stale walking. 5 Conclusion A new open loop controller has een presented which stailizes dynamic ounding walking in a class of simple quadruped roots. The strong gloal staility properties were illustrated via the step-to-step return map using numerical tools. Experimental implementation of this controller on two quadruped roots showed that, despite modeling errors, stale walking was achieved. Acknowledgments This project was supported in part y IRIS, a Federal Network of Centers of Excellence, the National Science and Engineering Research Council of Canada (NSERC), Terra Aerospace Corp., Aromat Inc, and Aacom Technologies. We also acknowledge the generous and talented technical support y D. McMordie and N. El-Fata. References [] R. Battaglia. Design and control of a four-legged root, SCOUT II. M. Eng. Thesis, McGill University, Mar 999. [] M. Buehler, R. Battaglia, A. Cocosco, G. Hawker, J. Sarkis, and K. Yamazaki. Scout: A simple quadruped that walks, clims and runs. In Proc. IEEE Int. Conf. Rootics and Automation, pages 77{7, Leuven, Belgium, May 998. [3] A. E. Cocosco. Control of walking in a quadruped root with sti legs. M. Eng. Thesis, McGill University, July 998. [4] E. R. Dunn and R. D. Howe. Foot placement and velocity control in smooth ipedal walking. In Proc. IEEE Int. Conf. Rootics and Automation, pages 578{ 583, Minneapolis, MN, Apr 996. [5] J. Furusho, A. Sano, M. Sakaguchi, and E. Koizumi. Realization of ounce gait in a quadruped root with articular-joint-type legs. In Proc. IEEE Int. Conf. Rootics and Automation, pages 697{7, May 995. [6] S. Hirose, K. Yoneda, K. Arai, and T. Ie. Design of a quadruped walking vehicle for dynamic walking and stair climing. Advanced Rootics, 9():7{4, 995. [7] T. McGeer. Passive dynamic walking. Int. J. Rootics Research, 9():6{8, 99. [8] H. Miura, I. Shimoyama, M. Mitsuishi, and H. Kimura. Dynamical walk of quadruped root (Collie-). In H. Hanafusa and H. Inoue, editors, Int. Symp. Rootics Research, pages 37{34. MIT Press, Camridge, 985. [9] K. Pearson. The control of walking. Scientic American, 35:7{86, 976. [] QNX Software Systems LTD. QNX Operating System v4.. Ontario, Canada, 996. [] M. H. Raiert. Legged Roots That Balance. MIT Press, Camridge, MA, 986. [] A. C. Smith and M. D. Berkemeier. Passive dynamic quadrupedal walking. In Proc. IEEE Int. Conf. Rootics and Automation, pages 34{39, Nantes, France, April 997. [3] K. S. Yamazaki. The design and control of SCOUT I, a simple quadruped root. M. Eng. Thesis, McGill University, Dec 998.

Using GPOPS-II to optimize sum of squared torques of a double pendulum as a prosthesis leg. Abstract

Using GPOPS-II to optimize sum of squared torques of a double pendulum as a prosthesis leg Abstract Milad Zarei MCE 593 Prosthesis Design & Control A two-dimensional, two links pendulum is developed to