Akihiro Sato's SLIP Hopper page

This page describes my Master's thesis research done in Ambulatory Robotics Lab, Centre for Intelligent Machines, McGill University. The topic is a biomechanically-inspired planar hopping robot named the SLIP Hopper. This project was supervised by Prof. Martin Buehler, who is now with Boston Dynamics.

Table of Contents:

Quick Overview

Motivation & Background

Dynamics Model

Control

Experimental Platform

Numerical Simulation Results (Data & Videos)

Experimental Results (Data & Videos)

Publication

Frequently Asked Questions

Highlights:

Simulation Video 1 (MPEG, 0.4 MB)

Experiment Video 1 (MPEG, 7.3 MB)

Experiment Video 2 (MPEG, 1.2 MB)

Quick Overview

The complex musculoskeletal system of a running animal on horizontal surfaces act essentially like a simple pogo stick. My research was focused on the development of a robotic pogo stick, i.e. a planar one-legged hopping robot with only one motor. A feasibility study was performed using numerical simulation. The experimental platform was designed and built based on biomechanics. Simulation and experimental data demonstrated periodic and robust stability. Running was achieved at 6.7 leg lengths per second, which is, to date, the fastest dimension-less speed for a single-legged robot.

More detailed Abstract (Résumé version Français)

Motivation & Background

- Robotic fast terrestrial mobility that can negotiate rough terrain is aimed at, i.e. running of a legged robot is studied;

- Inspired by biomechanics, in the last two decades, the Spring-loaded Inverted Pendulum (SLIP) model has been extensively used as a reduced-order model in analysis and control of running legged robots. However, the SLIP model itself has never been implemented in a robot and validated experimentally before;

- The simple SLIP architecture with a single actuator should be sufficient for the running motion on level surfaces only. This allows the focus on the fundamental of running motion.

- The use of only one actuator simplifies physical design and implementation. (In return, analysis and control can be more involved due to the resultant underactuation);

- SLIP Hopper is the first robot of its kind in terms of kinematics and the number of actuators as shown below:

Table: Comparison to previous work

Robot	Monopod	Bow Leg Hopper	Kenken	SLIP Hopper
(University)	(MIT)	(CMU)	(TITECH)	(McGill)
Kinematics	More complex than SLIP			SLIP
Models for analysis and control	SLIP			SLIP
# of actuator(s)	2			1

Dynamics Model

The SLIP model is used as the dynamics model in numerical simulation. The SLIP Hopper, the experimental robot, is designed based on the SLIP-model features and advantages.

Spring-loaded Inverted Pendulum (SLIP) model:

- Musculoskeletal systems of a running animal essentially act like a pogo stick. So do the mechanisms of a legged robot. The complicated mechanisms are simplified to a point-mass body with a springy leg to capture the approximate running dynamics in the sagittal plane;

- One degree of freedom (DOF) of leg rotation joint is actuated, and the other DOF of leg extension joint is unactuated with compliance (i.e. the system is underactuated). Only one control input is available for speed control and hopping height control;

- Entire dynamics model is a hybrid of two models of the flight and stance phases;

Flight dynamics is easily approximated by a ballistic trajectory, while stance dynamics is non-integrable and difficult to understand analytically;

- Equations of motion for the center of mass (COM) are derived for each of the flight and stance phases, using Lagurange equation.

Running Kangaroo and the SLIP model

Control

- PD controller is used to regulate leg angle via hip motor torque. This allows leg trajectories to have smooth changes in direction and velocity.

Hip motor torque generated by PD control regulates leg motion

- Phase-dependent switching control is used (the desired angles are switched at the phase transition events: touchdown and lift-off);

Diagram of implemented control code: mainly consisting of the desired angle switching and PD controller

Experimental Platform

Mechanical system structure:

- Robot has an actuated hip joint and a passive prismatic joint with a spring for bouncing (energy store/restitution);

- Motion of the robot body is constrained by a planar-constraint mechanism "planarizer" to approximate planar motion and to restrict pitch motion.

Isometric view of the CAD design of the robot with the planarizer

Actual robot with the planarizer

Computer & electrical system structure:

- Robot is tethered to the target computer via an interface card, which operates both D/A and A/D conversions;

- A control signal is sent from D/A to a motor amplifier, and sensor signals are acquired from A/D.

Numerical Simulation Results (Data & Videos)

Simulation Video 1 (MPEG, 0.4 MB), Presented in IROS 2004, etc.

- Start from a reasonable condition (initial condition in neighborhood to the steady state).

Simulation Video 2 (MPEG, 0.3 MB)

- Robust recovery from a disturbed condition (a negative forward speed at apex).

Phase plot of the vertical position of the COM:

- The combination of the two dynamics forms a closed orbit and repeats tracing the same pattern periodically, i.e. a limit cycle is generated;

- Motion on concern (i.e. vertical motion) is periodic at steady state.

Transition plot (plot of subsequent apex states):

- Apex state converges to the desired sate from an initial condition (IC) with a large error;

- Hopping motion would be robust to the disturbance that brings the robot state to a converged IC.

Region of attraction (collection of converged initial conditions in green):

- Region of attraction is large (i.e. robustness is high) without adaptive control;

- Convergence is assured in the neighborhood of steady state.

Experimental Results (Data & Videos)

Experiment Video 1 (MPEG, 7.3 MB)

Filmed Jun. 26, 2003. Presented in IROS 2004, etc.

- Infinite running starting from an initial condition.

Experiment Video 2 (MPEG, 1.2 MB)

Filmed Nov. 30, 2003. Published in the CD-ROM of IROS 2004.

- Close view of one cycle of steady running in high frame-rate video.

Snapshots from high frame-rate video:

- Dynamical state at one apex returns to the same state at the next apex;

- Continuous running from the infinite repetition of this gait cycle.

Hopping motion from one apex to the next

Phase plot of the vertical position of the COM:

- Limit cycle is generated, i.e. periodicity is demonstrated.

- Close correspondence to the phase plot with the simulation is shown.

Transition plot:

Apex state converges to the desired sate from an IC with a large error.

Region of attraction:

- Large region of attraction (i.e. high robustness) is indicated, based on 10 tested initial conditions.

Publication

Refereed international conference proceedings:

- A. Sato and M. Buehler, "A Planar Hopping Robot with One Actuator: Design, Simulation, and Experimental Results," IEEE/RSJ 17^th Int. Conf. on Intelligent Robots and Systems (IROS 2004), pp. 3540-3545, 2004.

- A. Sato, "Simulation of a one-legged hopping robot with phase plane stability," 18^th IASTED Int. Conf. on Modelling and Simulation (MS 2007), pp. 112-117, 2007.

- A. Sato, "Simple switching control for hybrid dynamics of a planar hopping robot," Ninth IASTED Int. Conf. on Control and Applications (CA 2007), pp. 63-68, 2007.

Frequently Asked Questions

Q1: What kind of robot is the SLIP Hopper?

A: The SLIP Hopper is a planar one-legged hopping robot. This robot is developed in order to study some principles of running dynamics that can be achieved by running legged robots. To simplify the analysis, only one leg is considered and only the COM motion in the Sagittal plane is considered. The word "planar" means that the robot can move in 2D, not in 3D, because the robot motion is constrained by a mechanism named the planarizer.

Q2: What is the main idea?

A: Many planar one-legged hopping robots in the past have two actuators. A popular arrangement is one actuator for the hip rotation to control forward speed and the other for the leg thrust to control hopping height. Those robots were successful, so it is a good idea that my research goes further. I aimed to control both forward speed and hopping height using only one actuator. The SLIP model is exploited and its passive spring in the leg is a key to allow it to run with one actuator. The results of my research have already shown that there exists a particular running gait that achieves a desired forward speed and a desired hopping height at the same time and that is very robust.

Q3: How can the robot run forward by using only one actuator? How can it jump upward without a thrust actuator in the telescopic leg?

A: The actuator is at the hip joint so that the robot can rotate the leg. This leg rotation propels the robot body pivoted at the ground contact of the leg toe so that it goes forward and upward. The leg also has a spring so that the robot does not have to push the body upward against the ground using a thrust actuator.

Q4: Is there any actuated joint in the planarizer?

A: No. All the joints in the planarizer are passive (in another word, unactuated). In the entire platform (the robot and planarizer), the only one actuated joint is the hip joint between the robot body and the robot leg.

Q5: Why is the body of the SLIP model a point-mass? Why is the leg massless? Isn't it unpractical?

A: You're right. Some hypotheses of the SLIP model are not practical. It is because the model was originally introduced to explain animal biomechanics a few decades ago and after that, it was developed by Raibert to analyze and control more complex robots than the SLIP model itself. The center of mass (COM) and the ground contact point of the foot were connected by an imaginary line, and the motion of the two points and the line were described using a spring. The SLIP model was originally not meant to be realized as an experimental platform.

Q6: Doesn't the hip torque spin the body? Should't the robot's pitch rotate?

A: No. In the ideal model, the body is a point mass (and has no moment of inertia as a result), and the leg is completely massless (and has no moment of inertia as a result). That means that the leg can be rotated to any angle you want without spinning the body at all. This theoretical model first appeared in the field of experimental biomechanics to explain animal locomotion, and the examination of the model was one of the research purposes here. Thus, no rotation of the body was desired.

In my simulation, the leg is massless, but its moment of inertia is considered so that I could check if the leg angle at touchdown can properly be set to a certain desired angle in time by using the limited performance of the motor installed on my experimental platform. One way to make it possible to simulate this is to set the inertia of the body to be big enough.

On the experimental platform, the center of mass is not located at the hip joint. However, the boom-planarizer constrains the pitch motion of the robot body so that we can assume that the center of mass coincides with the hip joint. Since the boom fixes the robot body, the hip torque doesn't spin the body.

Q7: Is it ok if posture control is not considered?

A: It is fine with the SLIP Hopper. The posture control of the body is not considered since the original SLIP model by definition has the center of mass coincident with the hip joint axis, i.e. the body did not have the moment of inertia or pitch motion to be considered. Thus, the posture angle is not defined. The SLIP Hopper is an experimental implementation of this model. Therefore, the robot has a constraint in its pitch motion.

I am aware that this constraint is not practical for applications to humanoid robots, for instance, but this constraint is just a result of sticking to the original SLIP model. Recently, some interesting papers on rigorous mathematical analysis of the SLIP model have been published and they take into consideration the offset between the hip joint and the center of mass and the pitch posture motion of the body part. To my knowledge, they haven’t shown the full properties of that kind of model. If it is possible to choose arbitrary hopping height and forward speed and the model is still stable, then the hopping height, the forward speed, and the body posture in the sagittal plane will be able to be controlled using only one actuator. I would like to have robots with such mechanical structures and controllers for more practical applications in the future.

Q8: Is the SLIP Hopper capable of hopping in place?

A: Unfortunately, no. The robot doesn't have any actuator to thrust, i.e. the leg cannot push off the ground in the direction of leg extension.

Q9: Did you do any simulation of vertical hopping in place?

A: No.

Q10: Is there any way to make the SLIP Hopper hop in place using a different controller?

A: Basically, no. It is not possible for the SLIP Hopper to keep hopping in place if the leg angle is fixed vertically. An external force has to be input vertically, but it can't be. The leg has to have some angle to the vertical and be rotated in the stance phase in order to have the vertical element of the ground reaction force created by a hip torque. As a result of this leg motion, the robot cannot stay in place. It might be possible to make it hop in place by using a certain combination of a leg angle and a hip torque that produces external forces such that the horizontal elements of the forces are all cancelled out.

Q11: Can the SLIP Hopper run without the boom-planarizer?

A: No. It is because the SLIP Hopper cannot stabilize lateral-plane motion. For now, pitch motion is not controlled either, but a pitch controller can be relatively easily implemented in the future since it is a matter of software, not hardware, if a control law is derived.

Q12: Do you think you can construct a robot that can hop around without a boom-planarizer by adding another motor at the hip joint to control lateral motion?

A: Yes. Actually, the Leg Lab at CMU/MIT constructed that kind of hopping robot two decades ago using two rotational actuators at the hip and a thrust actuator in the leg (three actuators in total). If the results of the SLIP Hopper are used, no thrust actuator will be needed (two actuators will be needed in total) unless vertical hopping is required.

Q13: Does the boom-planarizer have any counterbalanced mass on the opposite side of the boom? Does the planarizer have a bungee to lift up the robot or boom?

A: No, not at all.

Q14: The robot motion in the videos looks excellent. Is it reproducible and repeatable? Or are you showing the only trials that were successful?

A: The running motion was 100% repeatable without failure, as long as the initial condition was inside a reasonable range (i.e. inside the large region of attraction). For all the trials, the motion converged to a neighborhood of a desired steady-state within a few seconds.

Q15: How did you produce the initial conditions?

A: They were produced by releasing/throwing the robot in the air by hand. The first apex state is considered as the initial condition.

Q16: Why is there a coke can in the video of experiment?

A: Because it can show the size of the robot. It's not an advertisement.

If you have any comments/suggestions/other questions, please feel free to send an e-mail to me at .