Integration of Riemannian Motion Policy with Whole-Body Control for Collision-Free Legged Locomotion Daniel Marew1 Misha Lvovsky1 Shangqun Yu1 Shotaro Sessions1 and Donghyun Kim1

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Integration of Riemannian Motion Policy with Whole-Body Control for

Collision-Free Legged Locomotion

Daniel Marew 1, Misha Lvovsky1, Shangqun Yu1, Shotaro Sessions1, and Donghyun Kim1

Abstract— In this paper, we present a Riemannian Motion

Policy (RMP)ﬂow-based whole-body control framework for

improved dynamic legged locomotion. RMPﬂow is a differ-

ential geometry-inspired algorithm for fusing multiple task-

space policies (RMPs) into a conﬁguration space policy in a

geometrically consistent manner. RMP-based approaches are

especially suited for designing simultaneous tracking and colli-

sion avoidance behaviors and have been successfully deployed

on serial manipulators. However, one caveat of RMPﬂow is that

it is designed with fully actuated systems in mind. In this work,

we, for the ﬁrst time, extend it to the domain of dynamic-

legged systems, which have unforgiving under-actuation and

limited control input. Thorough push recovery experiments are

conducted in simulation to validate the overall framework.

We show that expanding the valid stepping region with an

RMP-based collision-avoidance swing leg controller improves

balance robustness against external disturbances by up to 53%

compared to a baseline approach using a restricted stepping

region. Furthermore, a point-foot biped robot is purpose-built

for experimental studies of dynamic biped locomotion. A pre-

liminary unassisted in-place stepping experiment is conducted

to show the viability of the control framework and hardware.

I. INTRODUCTION

A. Motivation

In dynamic legged locomotion, the size of valid stepping

region is crucial as it determines the maximum CoM velocity

that can be safely regulated [1]. Previous work on dynamic

legged locomotion conservatively restricted the stepping re-

gion in the lateral direction so that the robot’s legs do not

cross one another in order to mitigate the risk of self-collision

[2]–[4]. However, this restriction prevents the robots from

taking aggressive but stabilizing steps, thus reducing their

balance stability and robustness to external disturbances.

In this work, we aim to address this undue trade-off by de-

ploying an RMPﬂow-based [5] reactive collision-avoidance

swing leg controller. The controller steers the swing foot

towards the planned step location while avoiding collisions

between the robot’s links. This approach enables full uti-

lization of the kinematically reachable region as a valid

stepping area, thereby widening the robot’s stability margin

and improving its robustness to external disturbances.

B. Related Works

Collision avoidance algorithms can generally be divided

into two categories. The ﬁrst encompasses planning-based

approaches wherein long horizon collision-free trajectories

Authors are with the 1Manning College of Information and Com-

puter Sciences at University of Massachusetts Amherst, Amherst, MA,

140 Governors Dr, Amherst, MA 01002, USA. Corresponding Author:

robot.dhkim@gmail.com

(a) (b)

Embedded Computer

UP Xtreme

Knee, Hip Fl/Ex Module

AK80-9

Hip Ab/Ad Module

AK60-6

IMU

Vectornav VN-100

20 cm

22 cm

Virtal joints

Fig. 1. Point-foot biped robot, Pat. (a) Small scale point-foot biped with

6 actuated degrees of freedom. (b) Off-the-shelf components used to build

the robot.

are generated using either sampling-based motion plan-

ning techniques, such as Rapidly-Exploring Random Trees

(RRTs) [6]–[8], or optimization-based techniques [9]–[11].

Although these methods are commonly employed in robotic

manipulators, they tend to be computationally intensive,

rendering them less suitable for real-time applications, es-

pecially those involving high-speed movements as found in

dynamic legged locomotion.

The second strategy to tackle the collision avoidance

problem involves the application of reactive controllers.

Historically, Artiﬁcial Potential Field (APF)-based strategies

have been the predominant technique for designing reactive

collision avoidance behaviors. APF-based methods achieve

collision avoidance by applying a virtual repulsive force

to the robot to steer it away from obstacles [12], [13].

However, these techniques can pose calibration challenges,

often neglect full-body dynamics, and frequently produce

undesired behaviors due to local minima.

Another widely used reactive method is using a con-

strained optimization [14]–[16]. In this context, the recently

proposed Control Barrier Functions (CBFs) are particularly

noteworthy. Speciﬁcally, Khazoom et al. derived accelera-

tion level self-collision avoidance constraints using signed

distance-based CBFs and incorporated them into a whole-

body control QP formulation. However, these methods do

not tend to scale efﬁciently as the number of considered col-

lision elements increases. Each additional collision element

introduces a new constraint to the optimization problem,

potentially rendering these methods impractical for fast and

real-time applications. Moreover, conﬂicts between these

constraints can sometimes render the optimization problem

infeasible [16].

RMPﬂow [5] is a promising alternative to these approaches

arXiv:2210.03713v3 [cs.RO] 6 Nov 2023

because it scales well with the number of collision tasks

requiring a limited amount of additional computation power,

making it suitable for fast and real-time operations. More-

over, it mitigates local-minima-induced undesired behaviors

of APF-based methods by specifying the behavior with an

additional highly non-linear priority Riemannian metric that

stretches the space in the direction of obstacles [17]. This

priority metric heavily penalizes motion in the direction of

obstacles while being indifferent to motion in the orthogonal

direction [5]. In addition, because the priority metric can

be velocity dependent, it can be designed so that the robot

ignores nearby obstacles it is moving away from seamlessly

[17]. This behavior is crucial for high-speed operations and

cannot be achieved with APF methods that rely solely on

position-dependent repulsive forces.

In the RMPﬂow framework [5], [17], each desired task-

space behavior is speciﬁed by a local motion policy called

a Riemannian motion policy (RMP), which is composed on

the task space manifold. An RMP consists of an acceleration

policy and an associated Riemannian metric, also known as

a state-dependent inertia metric, that encodes information

about the task space’s geometry and the relative priority of

the task. The RMPﬂow algorithm fuses these local RMPs

into a single conﬁguration-space (C-space) RMP using an

operator called the pullback operator.

One caveat of RMPﬂow is its implicit assumption that the

C-space acceleration policy obtained through the pullback

operations is always realizable by the robot. This assumption

is often not satisﬁed when it comes to underactuated systems

like legged robots. To-date, the framework has been primarily

applied to fully actuated systems such as serial manipulators

[5], [17]–[20], with one notable exception [21]. Wingo et

al. [21] proposed an extension of RMPﬂow for a class of

underactuated wheeled inverted pendulum (WIP) robots. An

actuated joint directly controls the ﬂoating base of the class

of WIP robots considered in [21], [22]. The underactuation of

this class of robots only emanates from the fact that this joint

shares the same control torque as the wheelbase. This kind of

underactuation lends itself to the type of dynamics separation

scheme the RMPﬂow formulation in [21] relies on. This

underactuation, however, differs from the underactuation of

legged systems whose ﬂoating base can not be directly

controlled [22]. Thus the RMPﬂow framework developed in

[21] can not be directly applied to legged systems.

In this paper, we address this gap by integrating RMPﬂow

with a traditional null space projection-based whole-body

controller formulation so that tasks speciﬁed in terms of

RMPs can be realized on legged systems. In our approach,

the RMPs are executed in a way that is consistent with

robot’s contact-constrained dynamics while not compromis-

ing the tracking performance of higher priority tasks such as

ﬂoating base tasks. Speciﬁcally, we formulate a constrained

weighted least-squares problem inspired by the RMPﬂow

formulations in [5] and [21] whose optimal solution, a

C-space acceleration command, will try to realize RMPs

as faithfully as possible, giving priority according to the

assigned metric. Moreover, the solution is by construction

guaranteed not to violate contact constraints and constraints

derived from higher priority tasks.

Point-foot bipeds are relatively simple legged systems, but

since they are severely underactuated and highly unstable,

they are extremely challenging to control, making them an

ideal experimental platform for testing the limits of dynamic

biped locomotion. To that end, this paper presents a new

small point-foot biped named Pat designed and built to

experimentally validate the proposed whole-body framework.

In summary, the key contributions of our study include the

following:

1) We present a new formulation for integrating RMPﬂow

with a null-space projection-based whole-body con-

troller, enabling its application in legged robots.

2) Our extensive simulation experiments highlight that

our proposed collision-avoidance swing-leg controller,

developed based on the aforementioned formulation,

signiﬁcantly enhances the robustness of a point-foot

biped robot against external disturbances. In addition,

we apply and test high-speed self-collision avoidance

on a quadruped robot, and provide a comparative

analysis of our results against APF and CBF based

methods.

3) We designed a low cost point-foot biped robot for

the experimental study of dynamic biped locomotion.

The performance of our proposed controller and the

viability of the robot’s hardware is validated through

successful demonstrations of unassisted, in-place walk-

ing.

II. INTEGRATION OF RIEMANNIAN MOTION POLICY AND

WHOLE-BODY CONTROL

This section details the proposed approach for integrating

RMPﬂow with the traditional null-space projection-based

whole-body controller formulation. We use the following

general equations of motion of legged robots

A¨

q+b+g=S⊤

aτ+J⊤

cfr,(1)

where A∈Rn+6×n+6,b∈Rn+6, and g∈Rn+6, are

the generalized mass matrix, coriolis force, and gravitation

forces, respectively. Sa∈Rn+6×n+6 is the actuated joint

selection matrix. τ∈Rn+6,fr∈R3·nc, and Jc∈

R3·nc×n+6 are joint torque, augmented reaction force and

contact Jacobian, respectively. ¨

q∈R6+nis the conﬁguration

space acceleration where nthe number of actuated degrees

of freedom and ncis the number of contact points.

A. Review of RMPﬂow

In this section, we brieﬂy introduce the RMPﬂow compu-

tational framework ﬁrst proposed in [5]. RMP composed on

an m-dimensional manifold Mis characterized, in its canon-

ical form, by the tuple (a,M)M, where a:Rm×Rm→Rm

is an acceleration policy and M:Rm×Rm→Rm×m

+is a

state-dependent Riemannian metric. This tuple can be written

in its natural form, (f,M)M, by using f=Mawhere f

can be considered as a virtual force.

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IntegrationofRiemannianMotionPolicywithWhole-BodyControlforCollision-FreeLeggedLocomotionDanielMarew1,MishaLvovsky1,ShangqunYu1,ShotaroSessions1,andDonghyunKim1Abstract—Inthispaper,wepresentaRiemannianMotionPolicy(RMP)flow-basedwhole-bodycontrolframeworkforimproveddynamicleggedlocomotion.RMPflowisad...

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Integration of Riemannian Motion Policy with Whole-Body Control for Collision-Free Legged Locomotion Daniel Marew1 Misha Lvovsky1 Shangqun Yu1 Shotaro Sessions1 and Donghyun Kim1

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