Dynamic Locomotion in the MIT Cheetah 3 Through Convex Model-Predictive Control
Jared Di Carlo
1
, Patrick M. Wensing
2
, Benjamin Katz
3
, Gerardo Bledt
1,3
, and Sangbae Kim
3
Abstract— This paper presents an implementation of model
predictive control (MPC) to determine ground reaction forces
for a torque-controlled quadruped robot. The robot dynamics
are simplified to formulate the problem as convex optimization
while still capturing the full 3D nature of the system. With
the simplified model, ground reaction force planning problems
are formulated for prediction horizons of up to 0.5 seconds,
and are solved to optimality in under 1 ms at a rate of 20-30
Hz. Despite using a simplified model, the robot is capable of
robust locomotion at a variety of speeds. Experimental results
demonstrate control of gaits including stand, trot, flying-trot,
pronk, bound, pace, a 3-legged gait, and a full 3D gallop. The
robot achieved forward speeds of up to 3 m/s, lateral speeds up
to 1 m/s, and angular speeds up to 180 deg/sec. Our approach
is general enough to perform all these behaviors with the same
set of gains and weights.
I. INTRODUCTION
Control of highly dynamic legged robots is a challenging
problem due to the underactuation of the body during many
gaits and due to constraints placed on ground reaction forces.
As an example, during dynamic gaits
4
such as bounding or
galloping, the body of the robot is always underactuated.
Additionally, ground reaction forces must always remain in
a friction cone to avoid slipping. Current solutions for highly
dynamic locomotion include heuristic controllers for hopping
and bounding [1], which are effective, but difficult to tune;
two-dimensional planar simplifications [2], which are only
applicable for gaits without lateral or roll dynamics; and
evolutionary optimization for galloping [3], which cannot
currently be solved fast enough for online use. Recent
results on hardware include execution of bounding limit
cycles discovered offline with HyQ [4] and learned pronking,
trotting, and bounding gaits on StarlETH [5].
Predictive control can stabilize these dynamic gaits by
anticipating periods of flight or underactuation, but is dif-
ficult to solve due to the nonlinear dynamics of legged
robots and the large number of states and control inputs.
Nonlinear optimization has been shown to be effective for
predictive control of hopping robots [6], humanoids [7], [8],
and quadrupeds [9], with [9] demonstrating the utility of
heuristics to regularize the optimization. Another common
approach is to use both a high-level planner, such as in [10],
[11] and a lower level controller to track the plan. More
Authors are with the
1
Department of Electrical Engineering and
Computer Science at the Massachusetts Institute of Technology,
Cambridge, MA, 02139, USA; the
2
Department of Aerospace and
Mechanical Engineering at the University of Notre Dame, Notre
Dame, IN, 46556; and the
3
Department of Mechanical Engineer-
ing at the Massachusetts Institute of Technology, Cambridge, MA,
02139, USA. email: dicarloj@mit.edu, pwensing@nd.edu,
benkatz@mit.edu, gbledt@mit.edu, sangbae@mit.edu
4
In this paper, the term dynamic gaits is used to refer to gaits with
significant periods of flight or underactuation.
Fig. 1. The MIT Cheetah 3 Robot galloping at 2.5 m/s
recently, the experimental results in [12] show that whole-
body nonlinear MPC can be used to stabilize trotting and
jumping.
The stabilization of the quadruped robot HyQ using con-
vex optimization discussed in [13] demonstrates the utility
of convex optimization, but the approach cannot be imme-
diately extended to dynamic gaits due to the quasi-static
simplifications made to the robot model. Similarly, in bipedal
locomotion, convex optimization has been used to find the
best forces to satisfy instantaneous dynamics requirements
[14] and to plan footsteps with the linear inverted pendulum
model [15] but the latter approach does not include orienta-
tion in the predictive model.
While galloping is well studied in the field of biology
[16], [17], surprisingly few hardware implementations of
galloping exist. The first robot to demonstrate galloping
was the underactuated quadruped robot Scout II [18], which
reached 1.3 m/s, but had limited control of yaw. The MIT
Cheetah 1 robot [19] achieved high-speed galloping, but
was constrained to a plane. To the best of our knowledge,
the only previous implementation of a fully 3D gallop with
yaw control is on the hydraulically actuated WildCat robot
[20], developed by Boston Dynamics. Unfortunately, no
specific details about WildCat or its control system have been
published.
The main contribution of this paper is a predictive con-
troller which stabilizes a large number of gaits, including
those with complex orientation dynamics. On hardware,
we achieved a maximum yaw rate of 180 deg/sec and a
maximum linear velocity of 3.0 m/s during a fully 3D gallop,
which we believe to be the fastest gallop of an electrically
actuated robot, and the fastest angular velocity of any legged
robot similar in scale to Cheetah 3. Our controller can be
formulated as a single convex optimization problem which
considers a 3D, 12 DoF model of the robot. The solution of
This work was supported by the National Science Foundation [NSF-IIS-
1350879] and the Air Force Office of Scientific Research [FA2386-17-1-
4661]
2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
Madrid, Spain, October 1-5, 2018
978-1-5386-8094-0/18/$31.00 ©2018 IEEE 7440
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