Pranav A. Bhounsule

Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge

No legged walking robot yet approaches the high reliability and the low power usage of a walking person, even on flat ground. Here we describe a simple robot which makes small progress towards that goal. Ranger is a knee-less four-legged ‘bipedal’ robot which is energetically and computationally autonomous, except for radio controlled steering. Ranger walked 65.2 km in 186,076 steps in about 31 h without being touched by a human with a total cost of transport [TCOT ≡ P/mgv ] of 0.28, similar to human’s TCOT of ≈ 0.3. The high reliability and low energy use were achieved by: (a) development of an accurate bench-test-based simulation; (b) development of an intuitively tuned nominal trajectory based on simple locomotion models; and (c) offline design of a simple reflex-based (that is, event-driven discrete feed-forward) stabilizing controller. Further, once we replaced the intuitively tuned nominal trajectory with a trajectory found from numerical optimization, but still using event-based control, we could further reduce the TCOT to 0.19. At TCOT = 0.19, the robot’s total power of 11.5 W is used by sensors, processors and communications (45%), motor dissipation (≈34%) and positive mechanical work (≈21%). Ranger’s reliability and low energy use suggests that simplified implementation of offline trajectory optimization, stabilized by a low-bandwidth reflex-based controller, might lead to the energy-effective reliable walking of more complex robots.

Control of a compass gait walker based on energy regulation using ankle push-off and foot placement

In this paper, we present a theoretical study on the control of a compass gait walker using energy regulation between steps. We use a return map to relate the mid-stance robot kinetic energy between steps with two control inputs, namely, foot placement and ankle push-off. We show that by regulating robot kinetic energy between steps using the two control inputs, we are able to (1) generate a wide range of walking speeds and stride lengths, including average human walking; (2) cancel the effect of external disturbance fully in a single step (dead-beat control); and (3) switch from one periodic gait to another in a single step. We hope that insights from this control methodology can help develop robust controllers for practical bipedal robots.

Discrete-Decision Continuous-Actuation Control: Balance of an Inverted Pendulum and Pumping a Pendulum Swing

In some practical control problems of essentially-continuous systems, the goal is not to tightly track a trajectory in state space, but only some aspects of the state at various points along the trajectory, and possibly only loosely. Here we show examples in which classical discrete-control approaches can provide simple, low inputand low outputbandwidth control of such systems. The sensing occurs at discrete stateor time-based events. Based on the state at the event, we set a small set of control parameters. These parameters prescribe features, e.g. amplitudes of open-loop commands that, assuming perfect modeling, force the system to, or towards, goal points in the trajectory. Using this discrete decision continuous actuation (DDCA) control approach, we demonstrate stabilization of two examples: 1) linear dead-beat control of a time delayed linearized inverted pendulum; and 2) pumping of a hanging pendulum. Advantages of this approach include: It is computationally cheap compared to real-time control or online optimization; it can handle long time delays; it can fully correct disturbances in finite time (deadbeat control); it can be simple, using few control gains and set points and limited sensing; and it is low bandwidth for both sensing and actuator commands. We have found the approach useful for control of robotic walking.

Foot Placement in the Simplest Slope Walker Reveals a Wide Range of Walking Solutions

We show that the simplest slope walker can walk over wide combinations of step lengths and step velocities at a given ramp slope by proper choice of foot placement. We are able to find walking solutions up to slope of 15.42°, beyond which, the ground reaction force on the stance leg goes to zero, implying a flight phase. We also show that the simplest walker can walk at human-sized step length and step velocity at a slope of 6.62°. The central idea behind control using foot placement is to balance the potential energy gained during descent with the energy lost during collision at foot-strike. Finally, we give some suggestions on how the ideas from foot placement control and energy balance can be extended to realize walking motions on practical legged systems.

A Discrete Control Lyapunov Function for Exponential Orbital Stabilization of the Simplest Walker

In this paper, we demonstrate the application of a discrete control Lyapunov function (DCLF) for exponential orbital stabilization of the simplest walking model supplemented with an actuator between the legs. The Lyapunov function is defined as the square of the difference between the actual and nominal velocity of the un-actuated stance leg at the mid-stance position (stance leg is normal to the ramp). The foot placement is controlled to ensure an exponential decay in the Lyapunov function. In essence, DCLF does foot placement control to regulate the mid-stance walking velocity between successive steps. The DCLF is able to enlarge the basin of attraction by an order of magnitude and to increase the average number of steps to failure by two orders of magnitude over passive dynamic walking. We compare DCLF with a one-step dead-beat controller (full correction of disturbance in a single step) and find that both controllers have similar robustness. The one-step dead-beat controller provides the fastest convergence to the limit cycle while using least amount of energy per unit step. However, the one-step dead-beat controller is more sensitive to modeling errors. We also compare the DCLF with an eigenvalue-based controller for the same rate of convergence. Both controllers yield identical robustness but the DCLF is more energy-efficient and requires lower maximum torque. Our results suggest that the DCLF controller with moderate rate of convergence provides good compromise between robustness, energy-efficiency, and sensitivity to modeling errors.

Stable bipedal walking with a swing-leg protraction strategy

In bipedal locomotion, swing-leg protraction and retraction refer to the forward and backward motion, respectively, of the swing-leg before touchdown. Past studies have shown that swing-leg retraction strategy can lead to stable walking. We show that swing-leg protraction can also lead to stable walking. We use a simple 2D model of passive dynamic walking but with the addition of an actuator between the legs. We use the actuator to do full correction of the disturbance in a single step (a one-step dead-beat control). Specifically, for a given limit cycle we perturb the velocity at mid-stance. Then, we determine the foot placement strategy that allows the walker to return to the limit cycle in a single step. For a given limit cycle, we find that there is swing-leg protraction at shallow slopes and swing-leg retraction at steep slopes. As the limit cycle speed increases, the swing-leg protraction region increases. On close examination, we observe that the choice of swing-leg strategy is based on two opposing effects that determine the time from mid-stance to touchdown: the walker speed at mid-stance and the adjustment in the step length for one-step dead-beat control. When the walker speed dominates, the swing-leg retracts but when the step length dominates, the swing-leg protracts. This result suggests that swing-leg strategy for stable walking depends on the model parameters, the terrain, and the stability measure used for control. This novel finding has a clear implication in the development of controllers for robots, exoskeletons, and prosthetics and to understand stability in human gaits.

Dead-Beat Control of Walking for a Torso-Actuated Rimless Wheel Using an Event-Based, Discrete, Linear Controller

In this paper, we present dead-beat control of a torsoactuated rimless wheel model. We compute the steady state walking gait using a Poincaré map. When disturbed, this walking gait takes a few steps to cancel the effect of the disturbance but our goal is to develop a faster response. To do this, we develop an event-based, linear, discrete controller designed to cancel the effect of the disturbance in a single step – a one-step dead-beat controller. The controller uses the measured deviation of the stance leg velocity at mid-stance to set the torso angle to get the wheel back to the limit cycle at the following step. We show that this linear controller can correct for a height disturbance up to 3% leg length. The same controller can be used to transition from one walking speed to another in a single step. We make the modelbased controller insensitive to modeling errors by adding a small integral term allowing the robot to walk blindly on a 7o uphill incline and tolerate a 30% added mass. Finally, we report preliminary progress on a hardware prototype based on the model.

Iterative learning control for accurate task-space tracking with humanoid robots

Precise task-space tracking with manipulator-type systems requires accurate kinematics models. In contrast to traditional manipulators, it is difficult to obtain an accurate kinematic model of humanoid robots due to complex structure and link flexibility. Also, prolonged use of the robot will lead to some parts wearing out or being replaced with a slightly different alignment, thus throwing off the initial calibration. Therefore, there is a need to develop a control algorithm that can compensate for the modeling errors and quickly retune itself, if needed, taking into account the controller bandwidth limitations and high dimensionality of the system. In this paper, we develop an iterative learning control algorithm that can work with existing inverse kinematics solver to refine the joint-level control commands to enable precise tracking in the task space. We demonstrate the efficacy of the algorithm on a theme-park type humanoid that learns to track the figure eight in 18 trials and to serve a drink without spilling in 9 trials.

Iterative learning control for high-fidelity tracking of fast motions on entertainment humanoid robots

Creating animations for entertainment humanoid robots is a time consuming process because of high aesthetic quality requirements as well as poor tracking performance due to small actuators used in order to realize human size. Once deployed, such robots are also expected to work for years with minimum downtime for maintenance. In this paper, we demonstrate a successful application of an iterative learning control algorithm to automate the process of fine tuning choreographed human-speed motions on a 37 degree-of-freedom humanoid robot. By using good initial feed-forward commands generated by experimentally-identified joint models, the learning algorithm converges in about 9 iterations and achieves almost the same fidelity as the manually fine tuned motion.

Control Synergies for Rapid Stabilization and Enlarged Region of Attraction for a Model of Hopping

Inspired by biological control synergies, wherein fixed groups of muscles are activated in a coordinated fashion to perform tasks in a stable way, we present an analogous control approach for the stabilization of legged robots and apply it to a model of running. Our approach is based on the step-to-step notion of stability, also known as orbital stability, using an orbital control Lyapunov function. We map both the robot state at a suitably chosen Poincaré section (an instant in the locomotion cycle such as the mid-flight phase) and control actions (e.g., foot placement angle, thrust force, braking force) at the current step, to the robot state at the Poincaré section at the next step. This map is used to find the control action that leads to a steady state (nominal) gait. Next, we define a quadratic Lyapunov function at the Poincaré section. For a range of initial conditions, we find control actions that would minimize an energy metric while ensuring that the Lyapunov function decays exponentially fast between successive steps. For the model of running, we find that the optimization reveals three distinct control synergies depending on the initial conditions: (1) foot placement angle is used when total energy is the same as that of the steady state (nominal) gait; (2) foot placement angle and thrust force are used when total energy is less than the nominal; and (3) foot placement angle and braking force are used when total energy is more than the nominal.