STOMP: Stochastic Trajectory Optimization for Motion Planning
Mrinal Kalakrishnan1Sachin Chitta2Evangelos Theodorou1Peter Pastor1Stefan Schaal1
Abstract— We present a new approach to motion planning
using a stochastic trajectory optimization framework. The
approach relies on generating noisy trajectories to explore
the space around an initial (possibly infeasible) trajectory,
which are then combined to produced an updated trajectory
with lower cost. A cost function based on a combination of
obstacle and smoothness cost is optimized in each iteration. No
gradient information is required for the particular optimization
algorithm that we use and so general costs for which derivatives
may not be available (e.g. costs corresponding to constraints
and motor torques) can be included in the cost function. We
demonstrate the approach both in simulation and on a mobile
manipulation system for unconstrained and constrained tasks.
We experimentally show that the stochastic nature of STOMP
allows it to overcome local minima that gradient-based methods
like CHOMP can get stuck in.
Motion planning for manipulation and mobile systems has
received extensive interest in recent years. Motion planning
for avoiding collision has been the most common goal, but
there are other objectives like constraint handling, torque or
energy minimization and achieving smooth paths that may
be important in certain scenarios as well. Domestic and
retail scenarios, in particular, will have lots of cases where
constraint satisfaction may be a prime goal, e.g. carrying
a glass of water. Mobile manipulation systems may need
to minimize energy to conserve power and stay active for
a longer period of time. Non-smooth jerky trajectories can
cause actuator damage. There is deﬁnitely a need for a
motion planner that can address such scenarios.
In this paper, we present a new approach to motion
planning that can deal with general constraints. Our ap-
proach involves stochastic trajectory optimization using a
series of noisy trajectories. In each iteration, a series of
such trajectories is generated. The generated trajectories are
simulated to determine their costs which are then used to
update the candidate solution. Since no gradient information
is required in this process, general constraints and additional
non-smooth costs can be optimized.
We demonstrate our approach through both simulation
and experimental results with the PR2 mobile manipulation
robot. We consider end-effector orientation constraints, e.g.
constraints that require the end-effector to keep a grasped
object horizontal through the entire trajectory. We attempt
1Mrinal Kalakrishnan, Evangelos Theodorou, Peter
Pastor, and Stefan Schaal are with the CLMC Labora-
tory, University of Southern California, Los Angeles, USA
2Sachin Chitta is with Willow Garage Inc., Menlo Park, CA 94025,
Fig. 1. (a) The Willow Garage PR2 robot manipulating objects in a
household environment. (b) Simulation of the PR2 robot avoiding a pole
in a torque-optimal fashion.
to minimize collision costs using a signed distance ﬁeld
to measure proximity to obstacles. We further impose a
smoothness cost that results in trajectories that can be
directly executed on the robot without further smoothing.
We also demonstrate the optimization of motor torques used
to perform a movement.
II. REL ATED WORK
Sampling-based motion planning algorithms have proved
extremely successful in addressing manipulation prob-
lems , , , , . They have been used to address
task space constraints and to impose torque constraints while
lifting heavy objects . While sampling based planners
result in feasible plans, they are often lacking in the quality
of the paths produced. A secondary shortcutting step is often
required to smooth and improve the plans output by these
Another class of motion planners that have been applied
to manipulation systems are optimization based planners.
Motion planning for cooperating mobile manipulators was
addressed in  where an optimal control approach was used
to formulate the motion planning problem with obstacles
and constraints. The problem was solved using a numerical
approach by discretizing and solving for the desired plan.
In , a covariant gradient descent technique was used for
motion planning for a 7 DOF manipulator. This algorithm
(called CHOMP) minimized a combination of smoothness
and obstacle costs and required gradients for both. It used
a signed distance ﬁeld representation of the environment to
derive the gradients for obstacles in the environment. We
adopt a similar cost function for our approach, but in contrast
to CHOMP, our optimization approach can handle general
cost functions for which gradients are not available.
III. THE STOMP ALGORITHM
Traditionally, motion planning is deﬁned as the problem of
ﬁnding a collision-free trajectory from the start conﬁguration
to the goal conﬁguration. We treat motion planning as an
optimization problem, to search for a smooth trajectory that
minimizes costs corresponding to collisions and constraints.
Speciﬁcally, we consider trajectories of a ﬁxed duration T,
discretized into Nwaypoints, equally spaced in time. In
order to keep the notation simple, we ﬁrst derive the algo-
rithm for a 1-dimensional trajectory; this naturally extends
later to multiple dimensions. This 1-dimensional discretized
trajectory is represented as a vector θ∈RN. We assume
that the start and goal of the trajectory are given, and are
kept ﬁxed during the optimization process.
We now present an algorithm that iteratively optimizes
this discretized trajectory, subject to arbitrary state-dependent
costs. While we keep the cost function general in this section,
Section IV discusses our formulation for obstacle, constraint,
energy, and smoothness costs. We start with the following
θi) + 1
θ=N(θ,Σ)is a noisy parameter vector with
mean θand variance Σ.q(˜
θi)is an arbitrary state-dependent
cost function, which can include obstacle costs, constraints
and torques. Ris a positive semi-deﬁnite matrix representing
control costs. We choose Rsuch that θTRθrepresents the
sum of squared accelerations along the trajectory. Let A
be a ﬁnite differencing matrix that when multiplied by the
position vector θ, produces accelerations ¨
1 0 0 0 0 0
−210· · · 0 0 0
1−2 1 0 0 0
0 0 0 1 −2 1
0 0 0 · · · 0 1 −2
0 0 0 0 0 1
Thus, the selection of R=ATAensures that θTRθrepre-
sents the sum of squared accelerations along the trajectory.
Previous work  has demonstrated the optimization of
the non-stochastic version of Eqn. 1 using covariant func-
tional gradient descent techniques. In this work, we instead
optimize it using a derivative-free stochastic optimization
method. This allows us to optimize arbitrary costs q(˜
which derivatives are not available, or are non-differentiable
Taking the gradient of the expectation in Eqn. 1 with
respect to ˜
θ, we get:
θi) + 1
θ#!= 0 (5)
which leads to:
Further analysis results in:
The expression above can be written in the form E˜
δθGis now the gradient estimate deﬁned
Previous approaches  have used the analytical func-
tional gradient to derive an iterative gradient descent update
rule. While this may be efﬁcient, it requires a cost function
that is smooth and differentiable. Moreover, even though this
has not been proposed , for a given cost function J(θ)the
positive deﬁniteness condition of the hessian ∇θθJ(θ)>0
is required to guarantee convergence. Our proposed gradient
estimation is motivated by the limitations of gradient based
optimization when it comes to non-differentiable or non-
smooth cost functions. Inspired by previous work in the
probability matching literature  as well as recent work
in the areas of path integral reinforcement learning , we
propose an estimated gradient formulated as follows:
Essentially, the equation above is the expectation of δθ
(the noise in the parameter vector ˜
θ) under the probability
metric P= exp −1
θ)is the state de-
pendent cost deﬁned on the trajectory and it is designed as
θ) = hPN
θi)i. Thus the stochastic gradient is now
formulated as follows:
Even though our optimization procedure is static in the
sense that it does not require the execution of a trajectory
rollout, our gradient estimation process has strong connection
to how the gradient of the value function is computed in
the path integral stochastic optimal control framework .
More precisely the goal in the framework of stochastic
optimal control is to ﬁnd optimal controls that minimize
a performance criterion. In the case of the path integral
stochastic optimal control formalism, these controls are
computed for every state xti as δˆ
δuare the sampled controls and p(x)corresponds to the
5 10 15 20 25 30 35 40 45
1.2 x 10−4
5 10 15 20 25 30 35 40 45
Fig. 2. (a) Each curve depicts a column/row of the symmetric matrix R−1.
(b) 20 random samples of , drawn from a zero mean normal distribution
with covariance Σ=R−1.
THE STOMP ALGORITHM
–Start and goal positions x0and xN
–An initial 1-D discretized trajectory vector θ
–An state-dependent cost function q(θi)
–A=ﬁnite difference matrix (Eqn 2)
–M=R−1, with each column scaled such that the maximum
element is 1/N
•Repeat until convergence of trajectory cost Q(θ):
1) Create Knoisy trajectories, ˜
θKwith parameters θ+k,
2) For k= 1...K, compute:
θk,i) = q(˜
3) For i= 1...(N−1), compute: [˜
4) Compute δθ=M˜
5) Update θ←θ+δθ
6) Compute trajectory cost Q(θ) = PN
i=1 q(θi) + 1
probability of every trajectory τistarting from xti and
ending in the terminal state xtN . This probability is deﬁned
as p(x) = exp (−S(τi)) with S(τi)being the cost of the
path τi= (xti, ..., xtN ). Thus, p(x)is inversely proportional
to the cost S(τi)and therefore paths with higher cost will
have a lower contribution to the optimal controls than paths
with lower cost. The process above is repeated for every state
xti until the terminal xtN (multistage optimization). Then
the controls are updated according to u=u+δˆ
trajectories are generated. In our case, we assume that each
state cost q(θi)is purely dependent only on the parameter
θi, and we do not blame future or past costs on the current
state. Hence, we simplify the problem by deﬁning a local
trajectory cost S(θi) = q(θi), i.e., we remove cumulation of
costs. We ﬁnd that this simpliﬁcation signiﬁcantly accelerates
convergence in the experiments presented in Sec. V.
The ﬁnal update equations for STOMP are presented
in Table I. There are a few points which warrant further
In order to keep control costs of noisy trajectories low, we
sample the noise from a zero mean normal distribution,
with Σ=R−1as the covariance matrix, as shown in
Figure 2(b). This is preferable to sampling with Σ=I
for several reasons: (1) samples have a low control cost
TR, and thus allow exploration of the state space without
signiﬁcantly impacting the trajectory control cost; (2) these
noisy trajectories may be executed without trouble on a
physical system; (3) the samples do not cause the trajectory
to diverge from the start or goal. Goal-convergent exploration
is highly desirable in trajectory-based reinforcement learning
of point to point movements, where dynamical systems have
been designed that satisfy this property .
B. Trajectory updates
After generating Knoisy trajectories, we compute their
costs per time-step S(θk, i)(Table I, Step 2(a)). In Step
2(b), we compute the probabilties P(θk, i)of each noisy
trajectory, per time-step. The parameter λregulates the sen-
sitivity of the exponentiated cost, and can be automatically
optimized per time-step to maximally discriminate between
the experienced costs. We compute the exponential term in
Step 2(b) as:
λS(θk,i)=e−hS(θk,i )−min S(θk,i)
max S(θk,i)−min S(θk,i ), (11)
with hset to a constant, which we chose to be h= 10 in
all our evaluations. The max and min operators are over all
noisy trajectories k. The noisy update for each time-step is
then computed in Step 3 as the probability-weighted convex
combination of the noisy parameters for that time-step.
Finally, in Step 4, we smooth the noisy update using the
Mmatrix, before updating the trajectory parameters in Step
5. The Mmatrix is formed by scaling each column of R−1
(shown in Figure 2(a)) such that the highest element in the
column has a magnitude 1/N. This scaling ensures that no
updated parameter exceeds the range that was explored in
the noisy trajectories. Multiplication with Mensures that
the updated trajectory remains smooth, since it is essentially
a projection onto the basis vectors of R−1shown in Fig-
These trajectory updates can be considered safer than a
standard gradient descent update rule. The new trajectory is
essentially a convex combination of the noisy trajectories
which have already been evaluated, i.e. there are no un-
expected jumps to unexplored parts of the state space due
to a noisy gradient evaluation. Our iterative update rule is
analogous to an expectation-maximization (EM) algorithm,
in which we update the mean of our trajectory sampling
distribution to match the distribution of costs obtained from
sampling in the previous iteration. This procedure guarantees
that the average cost is non-increasing, if the sampling is
assumed to be dense , . An additional advantage is
that no gradient step-size parameter is required; the only
open parameter in this algorithm is the magnitude of the
IV. MOTION PLANNING FO R A ROBOT ARM
In this section, we discuss the application of the stochastic
trajectory optimization algorithm in Table I to the problem
of motion planning of a high-dimensional robot manipulator.
We address the design of a cost function that allows planning
for obstacle avoidance, optimization of task constraints, and
minimization of joint torques.
The algorithm in Table I was derived for a one-
dimensional discretized trajectory. Scaling this to multiple
dimensions simply involves performing the sampling and
update steps for each dimension independently in each
iteration. The computational complexity of the algorithm thus
scales linearly with the dimensionality of the problem. For
the application to robot motion planning, we represent the
trajectory in joint space, with a ﬁxed duration and discretiza-
tion. We assume that the start and goal conﬁgurations are
provided in joint space.
B. Cost Function
The cost function we use is comprised of obstacle costs
qo, constraint costs qc, and torque costs qt.
qo(θt) + qc(θt) + qt(θt)(12)
The additional smoothness cost θTRθis already incorpo-
rated in the optimization problem in Equation 1.
1) Obstacle costs: We use an obstacle cost function
similar to that used in previous work on optimization-
based motion planning . We start with a boolean voxel
representation of the environment, obtained either from a
laser scanner or from a triangle mesh model. Although our
algorithm can optimize such non-smooth boolean-valued cost
functions, faster convergence can be achieved by using a
function for which local gradients are available (or can be
estimated by sampling). We compute the signed Euclidean
Distance Transform (EDT)  of this voxel map. The
signed EDT d(x), computed throughout the voxel grid,
provides information about the distance to the boundary
of the closest obstacle, both inside and outside obstacles.
Values of the EDT are negative inside obstacles, zero at
the boundary, and positive outside obstacles. Thus, the EDT
provides discretized information about penetration depth,
contact and proximity.
We approximate the robot body Bas a set of overlapping
spheres b∈ B. We require all points in each sphere to be
a distance at least away from the closest obstacle. This
constraint can be simpliﬁed as the center of the sphere being
at least +raway from obstacles, where ris the sphere
radius. Thus, our obstacle cost function is as follows:
qo(θt) = X
where rbis the radius of sphere b,xbis the 3-D workspace
position of sphere bat time tas computed from the kinematic
model of the robot. A straightforward addition of cost
magnitudes over time would allow the robot to move through
a high-cost area very quickly in an attempt to lower the cost.
(a) (b) (c)
Fig. 3. (a) Simulation setup used to evaluate STOMP as a robot arm
motion planner. (b) Initial straight-line trajectory between two shelves.
(c) Trajectory optimized by STOMP to avoid collision with the shelf,
constrained to maintain the upright orientation of the gripper.
Multiplication of the cost by the magnitude of the workspace
velocity of the sphere (k˙xbk) avoids this effect .
2) Constraint costs: We optimize constraints on the end-
effector position and/or orientation by adding the magnitude
of constraint violations to the cost function.
qc(θt) = X
where Cis the set of all constraints, vcis a function that
computes the magnitude of constraint violation for constraint
3) Torque costs: Given a suitable dynamics model of
the robot, we can compute the feed-forward torque required
at each joint to track the desired trajectory using inverse
dynamics algorithms . The motor torques at every instant
of time are a function of the joint states and their derivatives:
where xt=θtrepresents the joint states at time t, and
xtare the joint velocities and accelerations respectively,
obtained by ﬁnite differentiation of θ.
Minimization of these torques can be achieved by adding
their magnitudes to the cost function:
C. Joint Limits
Joint limits can potentially be dealt with by adding a
term to the cost function that penalizes joint limit violations.
However, we prefer to eliminate them during the exploration
phase by clipping noisy trajectories θ+at the limits, i.e. an
L1projection on to the set of feasible values. Since the noisy
trajectory stays within the limits, the updated trajectory at
each iteration must also respect the limits, since it is a convex
combination of the noisy trajectories. Additionally, since the
convex combination of noise is smoothed through the M
matrix, the resulting updated trajectory smoothly touches the
joint limit as opposed to bumping into it at high speed.
V. EX PER IME NTS
STOMP is an algorithm that performs local optimization,
i.e. it ﬁnds a locally optimum trajectory rather than a global
one. Hence, performance will vary depending on the initial
RESULTS OBTAINED FROM PLANNING ARM TRAJECTORIES IN THE
SI MULATE D SHEL F ENVI RONM ENT SH OWN IN FIGURE 3( A).
Scenario STOMP CHOMP STOMP
Unconstrained Unconstrained Constrained
Number of 210 / 210 149 / 210 196 / 210
Avg. planning time 0.88 ±0.40 0.71 ±0.25 1.86 ±1.25
to success (sec)
Avg. iterations 52.1 ±26.6 167.1 ±113.8 110.1 ±78.0
trajectory used for optimization. STOMP cannot be expected
to solve typical motion planning benchmark problems like
the alpha puzzle  in a reasonable amount of time. In
this section, we evaluate the possibility of using STOMP
as a motion planner for a robot arm in typical day-to-day
tasks that might be expected of a manipulator robot. We
conduct experiments on a simulation of the Willow Garage
PR2 robot in a simulated world, followed by a demonstration
of performance on the real robot.
We created a simulation world containing a shelf with 15
cabinets, as shown in Figure 3(a). Seven of these cabinets
were reachable by the 7 degree-of-freedom right arm of the
PR2. We tested planning of arm trajectories between all
pairs of the reachable cabinets in both directions, resulting
in 42 different planning problems. Since the algorithm is
stochastic in nature, we repeated these experiments 5 times.
In each case, the initial trajectory used was a straight-line
through conﬁguration space of the robot. Each trajectory was
5 seconds long, discretized into 100 time-steps.
These planning problems were repeated in two scenarios
with two different cost functions. The unconstrained scenario
used a cost function which consisted of only obstacle costs
and smoothness costs. Success in this scenario implies the
generation of a collision-free trajectory. In the constrained
scenario, we added a constraint cost term to the cost function.
The task constraint was to keep the gripper upright at all
times, i.e. as though it were holding a glass of water.
Speciﬁcally, the roll and pitch of the gripper was constrained
to be within ±0.2 radians. Success in this scenario involves
generation of a collision-free trajectory that satisﬁes the task
constraint within the required tolerance. We also evaluated
the performance of CHOMP, a gradient-based optimizer 
on the unconstrained scenario using the same cost function.
The exploration noise magnitude for STOMP, and the gradi-
ent descent step size for CHOMP were both tuned to achieve
good performance without instability. Both algorithms were
capped at 500 iterations. For STOMP, K= 5 noisy trajectory
samples were generated at each iteration, and an additional
5 best samples from previous iterations used in the update
Table II shows the results obtained from these experi-
ments. STOMP produced a collision-free trajectory in all
(a) (b) (c)
Fig. 4. Planning problem used to evaluate torque minimization. (a) Plan
obtained without torque minimization: arm is stretched. (b,c) Two different
plans obtained with torque minimization. In (b), the entire arm is pulled
down, while in (c) the elbow is folded in: both solutions require lower
gravity compensation torques. Figure 5(b) shows the torques required for
trials in the unconstrained scenario. In contrast, CHOMP
fails on nearly 30% of them, presumably due to the gradient
descent getting stuck in local minima of the cost function1.
The execution times are comparable, even though CHOMP
usually requires more iterations to achieve success. This is
because each iteration of STOMP requires multiple trajectory
cost evaluations, but can make larger steps in a more stable
fashion than the CHOMP gradient update.
In the constrained scenario, 93.3% of trials resulted in
plans that were both collision-free and satisﬁed the task
constraints. Figure 5(a) shows the iterative evolution of
trajectory costs for one of the constrained planning problems,
averaged over ten trials.
These results are obtained when initializing the algorithm
with a na¨
ıve straight-line trajectory through conﬁguration
space, usually infeasible due to collisions with the envi-
ronment (see Fig. 3(b)). The algorithm is able to push the
trajectory out of collision with the environment, as well as
optimize secondary costs like constraints and torques. The
trajectory after optimization is ready for execution on a
robot, i.e., no smoothing is necessary as is commonly used
in conjunction with sampling-based planners .
In order to test the part of the cost function that deals with
minimization of torques, we ran 10 trials on the planning
problem shown in Figure 4, with and without the torque term
in the cost function. The resulting torques of the optimized
movements in both cases are shown in Figure 5(b). Since
the movements are rather slow, most of the torques are used
in gravity compensation. Hence the torques at the beginning
and end of the movement stay the same in both cases. The
planner ﬁnds parts of the state space towards the middle of
the movement which require lower torques to support the
weight of the arm. Interestingly, the planner usually ﬁnds
one of two solutions as shown in Figures 4(b) and 4(c): (1)
the entire arm is pulled down, or (2) the elbow is folded in on
itself, both of which require lower torque than the stretched
B. Real Robot
The attached video shows demonstrations of trajectories
planned using STOMP in a household environment, executed
1This result was obtained using the standard CHOMP gradient update
rule, not the Hamiltonian Monte Carlo variant. Please refer Sec. VI for
50 100 150 200
± 1 standard deviation
1 2 3 4
Sum of abs. joint torques (Nm)
No torque opt.
Fig. 5. (a) Iterative evolution of trajectory costs for 10 trials of STOMP on
a constrained planning task. (b) Feed-forward torques used in the planning
problem shown in Figure 4, with and without torque optimization, averaged
over 10 trials.
on the Willow Garage PR2 robot.
C. Code, Replication of Results
All of the software written for this work has been pub-
lished under the BSD open source license, and makes use of
the Robot Operating System (ROS) . Further instructions
on installing the software and replicating the results in this
paper can be found at .
A Hamiltonian Monte Carlo variant of CHOMP is dis-
cussed in , , as a principled way of introducing
stochasticity into the CHOMP gradient update rule. While
theoretically sound, we found this method difﬁcult to work
with in practice. It introduces additional parameters which
need to be tuned, and requires multiple random restarts
to obtain a successful solution. In contrast, our algorithm
requires minimal parameter tuning, does not need cost func-
tion gradients, and uses a stable update rule which under
certain assumptions guarantees that the average cost is non-
We have presented an algorithm for planning smooth
trajectories for high-dimensional robotic manipulators in the
presence of obstacles. The planner uses a derivative-free
stochastic optimization method to iteratively optimize cost
functions that may be non-differentiable and non-smooth. We
have demonstrated the algorithm both in simulation and on
a mobile manipulator, for obstacle avoidance, optimization
of task constraints and minimization of motor torques used
to execute the trajectory.
A possibility for future work is to augment this local
trajectory optimizer with a trajectory library approach, which
can recall previous trajectories used in similar situations, and
use them as a starting point for futher optimization .
The STOMP algorithm could also be applied to problems
in trajectory-based reinforcement learning, where costs can
only be measured by execution on a real system; we intend
to explore these avenues in future work.
ACK NOWL EDG EME NTS
This research was conducted while Mrinal Kalakrish-
nan and Peter Pastor were interns at Willow Garage. This
research was additionally supported in part by National
Science Foundation grants ECS-0326095, IIS-0535282,
IIS-1017134, CNS-0619937, IIS-0917318, CBET-0922784,
EECS-0926052, CNS-0960061, the DARPA program on
Advanced Robotic Manipulation, the Army Research Ofﬁce,
the Okawa Foundation, and the ATR Computational Neuro-
science Laboratories. Evangelos Theodorou was supported
by a Myronis Fellowship.
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