IROS2019国际学术会议论文集1259

Implementation of a Natural Dynamic Controller on an Under-actuated Compass-Biped Robot Ron Hartston, Rea Yakar, Reuven Katz andMiriam Zacksenhouse, member IEEE Sensory-Motor Integration Laboratory, Faculty of Mechanical Engineering, Technion - Israels Institute of Technology, Haifa 32000, Israel Email: hartstoncampus.technion.ac.il Email: mermzme.technion.ac.il AbstractNatural dynamic controllers aim to per- form the desired task by exploiting the natural dy- namics of the system. This can be accomplished by generating torque patterns to actuate the system rather than accurately following a predefined trajec- tory. We have previously demonstrated natural dy- namic control of compass-biped in simulations. Here we demonstrate successful implementation of this dynamic controller on an under-actuated compass- biped robot. The parameters of the controller, and in particular the magnitude and timing of torque primi- tives, were optimized using multi-objective optimiza- tion via genetic algorithm, accounting for speed and energy efficiency. While the current implementation is in open-loop, this strategy can be extended to include feedback to enhance walking over a wide range of terrains. This proof-of concept provides the basis for future extensions to more complex robots. I. Introduction Legged locomotion can be controlled by following optimized trajectories or by applying optimized torque patterns. Trajectory-based methods involve planning a trajectory for the configuration of the robot that satis- fies different artificial constraints to prevent falling. An additional control law is then used to provide precise trajectory tracking. The desired trajectory may be de- termined from foot-placement, as in quasi-static walking 1, and Zero Moment Point (ZMP) 2, 3, or from virtual constraints, as in Hybrid Zero Dynamics (HZD) 4. The reference trajectory tracking is enforced using high-gain feedback control and strong actuators, thus, suppressing the robots natural dynamics 5. This, leads to relatively poor energy efficiency. To improve energy efficiency, the desired trajectory may be derived from an energy efficient limit cycle. Interest in this approach arises from the work on Passive Dynamic Walkers (PDWs) 6, 7, 8. A PDW is a mech- anism, without any actuators, that exhibits periodic, limit-cycle walking when released down a slope. Adding actuation allows the robot to walk on flat terrains and even on upward slopes. In any case, trajectory-based methods require controllers to follow the desired limit cycle either continuously or discretely (i.e., once per step, as in the Cornell Ranger 9). In contrast, torque-based methods aim to generate orbital stable dynamical systems without explicit design of the limit cycle10. This approach facilitates exploiting the robots natural dynamics, and thus improving energy efficiency. The Cornell biped 11, 12, for example, is based on active ankle actuation, which is triggered by sensory feedback from switches. Limit-cycle walking can be generated by networks of coupled oscillators to coordinate either the desired trajectories or torques of the different joints. These networks have been suggested as viable models for bio- logical Central Patterns Generators (CPGs), i.e., neural circuits that coordinate muscle activation to generate rhythmic movements like swimming and walking 13, 14, 15, 16. The dynamics of the network provides intrinsic, albeit possibly limited, robustness to the gener- ated pattern of activity 17, 18, 19. Enhanced robust- ness and adaptation can be achieved by incorporating feedback and reflexes 20. The resulting robustness, as well as the ability to generate natural movements, motivated the development of artificial CPGs for robotic applications and in particular for walking robots 10, 21, 22, 18, 23. However, tuning the parameters of CPG networks to produce a desired movement is still a challenge. In our previous work we suggested a simplified 2- level CPG method to generate the pattern of torque activation for biped robots. The 2-level CPG is based on separate rhythm and pattern generators. In the simple case, the rhythm generator is just a single phase- oscillator, and the pattern generator produces a series of torque primitives at specific phases or events. A similar approach was used in 24 for open-loop actuation of the ankle of a hopper. Multi-objective optimization can be used to optimize the parameters of the rhythm and pattern generators for different desired objectives, and 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE2273 feedback can be added to modify those parameters to adapt to the environment 23. The purpose of this work is to demonstrate this approach, at least in open-loop, on an under-actuated robot. II. DYNAMIC CONTROLLER AND ITS OPTIMIZATION The dynamic controller generates torque signals to activate the robot and allows the natural dynamics of the robot to dictate the actual trajectory. A particu- lar implementation involves the generations of torque primitives, (e.g., torque pulses) at specific phases of an internal phase variable (t) 23: RG(t) = RGandRG(t) (mod1)(1) where RGand RG(t) are the frequency and phase of the rhythm generator. Fig. 1 depicts the activation of n = 2 torque pulses per a walking-step that were considered for actuating the hip of the compass-biped (CB) robot, described in this work, though in general more pulses can be used. While general torque primitives may be used, torque pulses have been shown to be advantageous compared to sinusoidal primitives, at least for hopping 24. However, to avoid impacts resulting from sharp torque changes it would be beneficial to consider smoothed torque pulses. Each pulse is defined by its magnitude i=1,.,nand the phases in which it is activated and deactivated, 2i1and 2i, respec- tively. When the other leg is the swing leg, the hip torque is applied in the opposite direction at the same phases. Additional parameters can be used to adapt the magnitude of the torque to the environment (as in 23), but at this stage we focus on open-loop control. Thus the controller considered here is specified by seven parameters RG,1,1,2,2,3,4. In order to clear the ground, the legs are extended and retracted at specific events during the walking cycle. In particular, the swing leg is extended when its angular velocity is zero, signaling that it reached its maximum height, and the stance leg is retracted after the swing leg hits the ground. The parameters of the controller were tuned using multi-objective optimization that accounted for two ob- jectives: (i) speed, and (ii) energy efficiency, as de- tailed in 23. The optimization was performed using multi-objective genetic algorithm (MOGA). Genetic al- gorithms, are stochastic directed-search algorithms mo- tivated by ideas from natural selection and evolution 25, 26. In each generation 15% the most dominant solutions (i.e., solutions that are on the outer-most Pareto fronts) were selected for the next generation. The next generation was augmented by generating additional solutions via mutation and reproduction, as detailed in 23. Fig. 1.Torque () pulses are activated and terminated at par- ticular phases of an internal phase variable (). The timing of leg extension and contraction are marked for illustration but are activated by event-driven triggers. (a) Robot diagram(b) The robot Fig. 2.Compass-Biped robot. Left: Robot diagram showing the central leg (blue), the outer legs (green), the rotor (red), the central-leg to rotor angle (m), and rotor to outer-leg angle (L). See also Fig. 3. Right: The robot. III. ROBOT IMPLEMENTATION A. Compass-biped Robot: Hardware The CB robot, shown in Fig. 2, includes a central leg and two outer legs. The two outer legs are rigidly attached so they function as one leg while increasing lateral stability. The robot is activated only at the hip via a torsional series elastic actuator (SEA) 27, 28, 29, as further detailed in Fig. 3. The housing of the motor of the SEA is attached to the central leg, and is loaded, via a torsional spring, by the two outer legs. The torque generated by the SEA (SEA) is related to the rotor to outer-leg angle (L) see Fig. 2, according to Eq. (2) SEA= KSEAL+ cSEAL(2) 2274 where KSEAand cSEAare the stiffness and viscous friction of the SEA, respectively (Table I). Actuation is limited by the gear/motor to 1Nm torque at 130rpm. Leg extension and retraction is accomplished via cam mechanisms at the end of each leg, as can be seen in the picture of the robot in Fig. 2. A micro processor board is attached to the central leg near the hip joint. This control board does all the real time processing (i.e. encoder sampling and control). The CB is completely autonomous, other than 12V power supply and UART connection for data retrieval. Fig. 3. Series Elastic Actuation (SEA). Motor housing is attached to the central leg (blue). The rotor (red) drives the outer legs (green) through a couple of torsional springs (grey). See also Fig. 2. B. System modeling and simulations The robot can be modeled as the well-studied compass-gait biped 3023. The compass gait can be divided into two phases: (I) a continuous, single-support phase where only one leg is in contact with the ground and (II) a discrete, double-support phase that occurs when the swing leg impacts the ground and becomes the new stance leg. Each leg has a mass m length L and moment of inertia Ihip(See Table I). The legs are connected at the hip, where an additional mass M is located. The leg in contact with the ground is referred to as the stance leg, while the other leg is referred to as the swing leg. Simulations are based on the dynamic equations detailed in 23 and 30. The parameters of the robot were identified during experiments that were conducted with the central leg attached to a rigid frame. In this case, the outer-legs behave as a forced pendulum, so the dynamics of the inter-leg angle = m Lis governed by Eq. (3) where SEA, the torque applied by the SEA, is given by Eq. (2) Ihip + chip + mgLsin = SEA(3) The identified parameters are listed in Table I. The gear/motor was modeled as well, although this model and its results are beyond the scope of this paper. TABLE I Compass Biped parameters obtained by system identification ParametersSymbolValueUnits Leg inertia w.r.t. hip axisIhip0.069kg m2 Hip axis viscous frictionchip0.01Nmsec Leg center of gravity w.r.t. hip axis a0.31m Leg total length from ankle to hip (Stance/Swing) L0.56/0.55m Leg weight including footm0.5kg Hip joint massM0.84kg Gravityg9.81m/s2 SEA stiffnessKSEA0.55Nm/rad SEA viscous frictionCSEA0.1Nmsec C. Controller tuning via MOGA The parameters of the controller RG,1,1,2, were optimized via MOGA, using a population of 2000 items, over 20 generations. The controller was optimized for a single pulse. The 1st, 7thand 15thPareto fronts in the last (20th) generation are depicted in Fig. 4, showing 00.811.21.4 Speed fitness 0.75 0.8 0.85 0.9 0.95 1 Energy fitness front #1 front #7 front #15 Selected controller Fig. 4. 1st, 7thand 15thPareto fronts in the speed fitness (x-axis) and energy efficiency fitness (y-axis) plane of the final population. The performance of the controller that was selected as a basis for implementation on the robot is depicted as a green star the trade-off between speed and energy efficiency for optimized controllers. The performance of the controller that was selected as a basis for implementation on the robot is depicted as a green star. The controller generates a strong torque pulse that throws the leg forward in the beginning of the swing phase as specified in Table II and depicted in Fig. 5. Extending the controllers to include two pulses, the MOGA resulted in a similar pulse in the beginning of the step, followed by lower, negative, torque pulse that slows down the swing leg before hitting the ground. 2275 TABLE II Controller parameters ParameterUnitsSimulation value Actual value Symbol Phase frequencyHz1.161.21RG 1st Hip pulse amplitude Nm10.851 1st Hip pulse activation phase0.010.011 1st Hip pulse deactivation phase0.230.252 IV. Results A. Simulation Results The simulated dynamics of the compass-biped robot with the selected controller parameters (Fig. 4, and third column of Table II) are evaluated for comparison with the known dynamics in the literature and with the ex- perimental results detailed in the next sub-section. Fig. 5 depicts the resulting inter-leg angle as a function of time during stable walking. Fig. 6 describes the dynamics of the simulated robot in the phase plane defined by the angular position and angular velocity of the leg ( ). It demonstrates the typical trajectories of the CB in this plane 30, 23, including the large arc during the swing phase (red), the inverted-V shaped trajectory during the stance phase (blue) and the jump between them due to the impact with the ground. Fig. 5.Simulation results: Torque pulses corresponding to the selected controller (see Fig. 4) and the resulting inter-leg angle (red). B. Experimental Results Initial hardware implementation is based on open- loop control in which the control signal is the voltage to the motor. The torque generated by the motor was determined from a model of the motor as depicted in Fig. 6. Simulation results: The dynamics of the simulated robot in the phase plane defined by the angular position and angular velocity of the leg ( ) during swing (red) and stance (blue) phases. Fig. 7. The motor and hip encoders were sampled at 100Hz and used online to trigger leg extension and retraction. Stable walking was achieved successfully, as demonstrated in the video (supplementary material), in Fig. 7 and Fig. 8. Fig. 7 indicates that stable walking was achieved. Fig. 8 describes the dynamics of the robot in the phase plane defined by the inter-leg angular position and velocity ( ), and compares the simulated and measured tra- jectories. The measured trajectories are based on offline filtering of the encoder readings (using a 5thorder zero- phase infinite impulse response filter with cutoff fre- quency of 30Hz). The shape of the measured trajectories is similar to the shape of the simulated trajectories. The magnitude of the inter-leg angle and its angular velocity are larger during the experiments, possibly due to the differences in the implementation of the controller and modeling differences. The experimental trajectories are slightly asymmetrical, possibly due to inertial differences between the central leg and outer legs. The average robot speed during the limit cycle walk- ing was deduced from the encoder reading by taking half of the average absolute value of the angular velocity resulting in 1.2m/s. This result was validated by esti- mating the mean velocity from the time it took to travel a specific distance. As previously mentioned, this result is higher than average speed calculated in simulation, 0.7m/s. Hip joint mechanical energy, during limit cycle, was estimated by calculating the applied torque and motor speed. The motor was activated for 0.15sec and its average speed during that time was 3.5rad sec, as calcu- lated from the measured motor angle. Motor torque was 2276 estimated from motor simulations to be 0.85Nm. Thus, the 12 steps taken in Fig. 7 required mechanical energy of Emech= 0.853.50.1512 = 5.5Joul. Those 12 steps covered a distance of 6.5m, so, given that the mass of the robot is 1.9kg, the estimated cost of transport (COT) is: COTmech= 5.5 1.9 6.5 g = 0.05 This result is very similar to the mechanical COT achieved by similar robots 11,28 as expected from a dynamic walker. The efficiency fitness calculated from this result, using the same method detailed in 23, yields 0.8. This is similar to simulated robot performance. Fig. 7.Experimental results: Torque generated by motor (blue) and inter-leg hip angle (red). Fig. 8.Experimental (blue) and simulation(green) results: The dynamics of the simulated robot in the phase plane defined by the inter-leg angle and angular velocity ( ). V. Conclusions We demonstrated a successful implementation of our strategy for generating limit-cycle walking that exploits the natural dynamics of the robot, rather than imposing prescribed trajectories. Optimization with MOGA was performed in simulations, as a basis for selecting proper parameters. The controller is driving the robot to a stable gait, even though it is currently implemented in open loop. Near term activity will close the loop on the applied torque by closing the loop on the rotor to outer-legs an- gle. However, to avoid the application of sharp changes in the torque, a smoother version of the pulse primitives will be considered. Future activity will be aimed at in- creasing the robustness of the gait to perturbations and terrain conditions, by integrating feedback to modify the controller parameters, as suggested in our previous work23. Acknowlegment This work was supported by the Technions Au- tonomous Systems Program, under grants # 2021775 and Gordon Center for System Engineering Grants #2023426 and #2026779. We also thank Dr. Jonathan Spitz for initiating and motivating this project, Mr. Shamsutdinov Rakhmatulla, the lab engineer, and the undergraduate students, Nadav Angel and Gal Barkai, for technical support. References 1 E. Rimon, S. Shoval, and A. Shapiro, “Design of a quadruped robot for motion with quasistatic force constraints,” Au- tonomous Robots, vol. 10, no. 3, pp. 279296, 2001. 2 K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka, “The development of honda humanoid robot,” in Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No. 98CH36146), vol. 2.IE