I Introduction
Optimization of robot postures and trajectories is an important problem in robotics, with applications to efficient robot locomotion and manipulation. The problem is hard because posture and trajectory optimization problems are often nonconvex and have several local optima, especially in the presence of collision constraints. For example when a robot model is in collision, the collision gradient might pull different links in different directions  and the optimization get stuck at an infeasible local optima [1]. While several algorithms have been proposed for posture generation and trajectory optimization [2, 3, 1, 4, 5, 6, 7], comparing them is also difficult since each researcher opts for a different evaluation scenario which was chosen by chance or to showcase the advantages of the algorithm.
This paper is a step towards solving these two issues: 1) of the lack of a benchmark, and 2) of avoiding infeasible local optima. The contributions of the paper are as follows.

We develop a new robotics challenge, open to the public, which consists of a set of problems for benchmarking static posture and trajectory optimization algorithms on a legged humanoid robot

We evaluate different optimization algorithms at solving the problems

We show that new stochastic optimization algorithms developed for the (highly nonconvex) training of deep neural networks also lead to high success rates in posture/trajectory optimization with tough collision constraints and initializations. To the best of our knowledge this is the first time these methods are applied to posture generation and trajectory optimization.

We propose a stochastic incremental variant of Sequential Quadratic Programming for trajectory optimization based on the principles of recent largescale nonconvex optimization methods. We show it leads to higher success rates and speed than SQP on illinitialized trajectory optimization problems.
Ii Related Work
Robot motion planning has been tackled with search, sampling and optimization methods. Recently, optimization algorithms have gained popularity, due to the existence of fast generalpurpose software and the possibility to easily integrate many different constraints in the problem. One of the stateoftheart algorithms is sequential quadratic programming, which is used by SNOPT [8] for generalpurpose optimization, but also by trajectoryoptimization libraries [1], and trajectory optimization research on legged robots [3]. Projected conjugategradient methods such as CHOMP [4] have also been proposed for the problem. The gradientdescent methods we evaluate in this paper are related to CHOMP in the sense that they also do (preconditioned) gradient descent. However, as in [1] we use penalties for constraints, thus being able to consider general nonlinear constraints on robot postures or motion.
In this paper we explore the use of stochastic methods for posture generation and trajectory optimization. The motivation behind it is to improve success rates by successfully avoiding local minima through random perturbations. The idea has also been explored in stochastic variants of CHOMP [5]
which increased success rates. Here we instead make use of progress in the stochastic optimization literature, which has recently gained attention in part because of the problem of local minima, saddle points and nonconvexities which pervade deep neural network training landscapes. The largescale optimization and deep learning communities have recently come up with different algorithms to deal with these optimization landscapes, such as variants of stochastic gradient descent with preconditioning
[9], incremental gradient descent methods [10], noise injection [11] and others. Some of these algorithms have provable convergence guarantees [10] and saddleescaping guarantees [12]. Our assumption in this paper is that these methods and insights which work on the highlynonconvex optimization landscapes of neural networks will transfer to the (also nonconvex) landscapes of legged robot posture generation and trajectory optimization.Results of stateoftheart robot motion planning algorithms are impressive [3, 2, 13]
, but it is arguably difficult to compare each planner’s performance, advantages and disadvantages. This is partly because each algorithm is evaluated on a different environment, or using different cost functions, constraints or robot models. For results to be comparable and verifiable, the evaluation criteria must be the same and largely sampled, while all inputs (i.e. scenario, robot, constraints) must be available. Recently, verifiability and comparability have been strongly pursued in fields such as computer vision through an investment in open benchmarks
[14, 15] and open source  which has arguably been a strong factor in fostering research progress. This paper tries to follow this trend and make public a benchmark with predefined robot, environment, costs, constraints and evaluation criteria.Iii THE LegOpt BENCHMARK
Iiia The environment
The LegOpt Benchmark (Leggedrobot posture and trajectory Optimization Benchmark), which we propose in this paper, is based on the “Destruction Scenario Dataset” (DSD)^{1}^{1}1The first author of this paper collaborated with Roboptics, Inc. on the development of the Destruction Scenarios Dataset. [16]. DSD is a set of challenging 3D scenarios intended for testing robot search and rescue missions. They were created in Blender using simple shape primitives, unions and differences, followed by shattering and collapsing by running a physics simulation. The scenarios also include 3D models of cars and people for realism. Snapshots of the first frames were taken as different “difficulty levels” of the scenarios according to an empirical measure of difficulty of traversing the terrain. Models and code to generate them are openly available^{2}^{2}2https://github.com/roboptics/destruction_scenarios.
The current version of the LegOpt Benchmark is built on top of the easiest difficulty garage scenario (i.e. “garage_easier”). We show an image of the scenario in Figure 1. This scenario is already challenging enough for current motion planning and optimization algorithms.
IiiB Task definitions
The LegOpt Benchmark consists of two tasks: posture generation and trajectory optimization. Each task is evaluated in 50 different problem instances (i.e. different regions of the scenario). Each posture generation problem is specified by a single stance with given 6D poses for the robot’s feet. Each trajectory optimization problem is specified by a sequence of stances with given feet poses.
Both tasks use Atlas as the robot model and DSD’s “garage_easier” scenario as the environment. The following constraints need to be satisfied for both tasks:

Given feet positions and orientations must be respected,

The centerofmass (COM) of the robot should lie inside the support polygon, given by a 2D convex hull of the horizontal projection of the feet in contact,

The robot model should not collide with itself or the environment,

Robot joint angle limits should be respected.
Each of these constraints is to be enforced with tolerance (meters for distances, radians for angles). We compute collision as the penetration distance in meters, as implemented in the OpenRAVE library [17] using ODE [18] for collision computations^{3}^{3}3We ignore collisions between the pelvis and uppertorso, since their convex hulls are always in collision and most collision checkers make use of convex hulls.. We provide an evaluation script which computes the costs and constraint violations of JSONformatted solutions on a website^{4}^{4}4https://github.com/martimbrandao/legoptbenchmark. We also provide JSON files describing the feet poses of each problem instance.
We evaluate an algorithm’s performance at the tasks using the following criteria: 1) Success rate, i.e. number of problem instances solved to tolerance; 2) Cost; and 3) Computation time. The cost function is different for the two tasks.
IiiB1 Posture generation
The posture generation task is to obtain a single fullbody configuration that respects given feet poses, static stability, nocollision, joint limits, and leads to the minimum (squared) static torques possible. Basically, the task is to solve
(1)  
s.t.  (2)  
(3)  
(4)  
(5) 
where is the number of the robot’s joints, are the problem’s given feet poses, is the convex support polygon, represents a signeddistance function between the robot, itself and the environment.
IiiB2 Trajectory optimization
The trajectory optimization task is to obtain a sequence of fullbody configurations (two configurations per stance) which respects given feet poses, static stability, nocollision, joint limits, and leads to the minimum (squared) joint velocity possible. Basically, the task is to solve
(6)  
s.t.  (7)  
(8)  
(9)  
(10) 
IiiC Generation of problem instances
For completeness we will now describe the procedure we used to arrive at the definition of all problem instances. We first sampled points uniformly from the scenario’s mesh triangles to obtain positions for a left foot, then we uniformly sampled yaw orientations, which together with the mesh’s normal vector make an orientation constraint. We then searched for a right foot position and orientation in the same way, in a radius around the left foot. We checked for feasibility and collision of the stance using TrajOpt. This was done until we had 100 stances scattered throughout the scenario. We then selected random pairs of stances (50 pairs on the same floor, 50 on different floors), and ran an ARA*based
[19] footstep planner [20]to obtain a sequence of stances which connects the pair. The sequence of stances from the first 25 successful samefloor footstep plans, and the first 25 successful differentfloor footstep plans were selected as the problem instances for trajectory optimization. Regarding posture generation, we used Kmeans clustering to pick 50 stances out of the set of all stances in all footstep plans (using stance length, stance height and trunk collision distance as features).
We show the distribution of problem definition features in Figure 2 and 3. The collision distance feature is computed by a signeddistance function between the trunk (approximated by a box) and the environment. Notice that many postures in both tasks are close to collision, and some will actually be in collision depending on the initialization  since only the trunk and not the arms were checked for collision while building the problem definitions.
Iv Benchmarked Algorithms
Posture generation and motion planning can be formulated as optimization problems of the form
(11a)  
s.t.  (11b)  
(11c)  
(11d) 
where are the fullbody postures of the robot at one or more instants of time (i.e. where
is the number of the degrees of freedom of the robot and
the number of instants of time a.k.a. waypoints). Each , , is a function which describes part of the costs or constraints on the robot, for example a fullbody posture cost or a link pose constraint at a certain waypoint. In this paper we turn (11) into a sequence of problems of the form(12a)  
s.t.  (12b) 
which are solved with successively higher until all constraints are satisfied (or progress is no longer possible). The functions and are called penalty functions and measure constraint violation. We use and , which are exact  meaning that they can lead to the exact solution to the original problem (11) for certain choices of [21].
Iva Sequential quadratic programming (SQP)
One of the stateoftheart methods for solving (11) is TrajOpt [1]. The method uses sequential quadratic programming, by alternating between

building convexifications , , , where are quadratic and , are linear

solving (12) with the convexified , , instead of the original , , functions using a trustregion method (i.e. constraining at each iteration to be within a narrow range where convexification is valid). TrajOpt also uses and
An implementation of this algorithm is openly available^{5}^{5}5http://rll.berkeley.edu/trajopt/ and achieves both a high success rate and computation speed when compared to other recent optimizationbased planners [4] and samplingbased planners.
IvB Gradient descent
A simpler way to solve (12) is to use gradient descent. For convenience, let us simplify the notation of the optimization problem (12) as
(13) 
where , and where the terms gather all cost functions and penalties in (12a):
(14) 
Gradient descent solves (13) by iterating
(15) 
where projects points onto the feasible space , and is called the step length. The step length is usually fixed or computed through line search [21]. The iterations are repeated until some termination criteria is met, usually a threshold on gradient norm or function improvement. Notice that the gradient of and is not defined at , and so computing (15) with these penalty functions is not straightforward. In this paper we use smooth approximators^{6}^{6}6Another option is to use slack variables [21] as done in TrajOpt [1]. We chose to use smooth approximators as it made debugging more intuitive. for the penalty functions such that:
(16a)  
(16b) 
Note that as , (16a) tends to for and tends to for , thus approximating the gradient of the exact penalty ; and similarly (16b) also approximates the gradient of .
IvC Popular algorithms in largescale optimization and deep learning
IvC1 Stochastic gradient descent
Stochastic gradient descent (SGD) is gradient descent where the gradients are noisy. In this situation an estimator
is used that is equal to the real gradient in expectation, i.e. . A common choice for this estimator is the use of minibatches, which basically evaluates the gradient at only a subset of the functions at each iteration. So the sum over is made not for , but only for a subset of those indices chosen randomly at iteration . An SGD iteration to solve (13) then consists of(17) 
The size of a minibatch () is usually called “samplesize”, and for SGD reduces to full deterministic gradient descent (15
). SGD has low complexity (i.e. low total number of gradient evaluations) and provable convergence in probability even on nonconvex functions
[22], while its noise actually helps avoid sharp local minima [23]. Due to these advantages the method has inspired the development of a number of different variants, which try to deal with important disadvantages such as slow convergence and hardtotune step lengths.IvC2 Adam
In this paper we will consider one of those variants which has become widely popular in the deep learning community: Adam [9]. Adam basically does SGD where the step length is adapted to each dimension of , similarly in purpose to conjugate gradient descent. In a nutshell, Adam iterates:
(18) 
where the division is elementwise, and ,
are unbiased running averages of the first and second moments of
, accumulated at each iteration the following way(19a)  
(19b)  
(19c)  
(19d)  
(19e) 
The constants and are parameters. We refer the interested reader to the original publication for details.
IvC3 Nadam
We also consider an accelerated version of Adam, Nadam [24]
, which uses Nesterov momentum in an attempt to increase convergence rate. The algorithm uses a slightly different definition for
,(20) 
IvC4 Incremental SGD (a.k.a. stochastic average gradient)
Stochastic average gradient, proposed in [10], is an incremental variant of SGD. The gradients of all functions are used at each iteration, but only a subset of them is actually computed: the rest are kept unchanged from the previous iteration. The method has faster convergence properties in convex functions than SGD, at the cost of increased memory usage. It iterates
(21) 
where
(22) 
As before, the set is chosen randomly at each iteration.
In the experimental section we will also consider incremental versions of Adam and Nadam (i.e. by keeping previous gradients in memory). For a shared and readable naming, we will refer to all these variants by the prefix ”I”, for incremental: ISGD, IAdam, INadam.^{7}^{7}7Note that ISGD and stochastic average gradient refer to the same method in this paper.
IvD A new, incremental SQP (ISQP)
In the spirit of ISGD, we propose to use an incremental version of SQP. The motivation is to observe that a large part of the computational time of SQP goes into building convexifications (i.e. basically taking the gradients of , , ), and so convergence rate can hypothetically be increased by reusing them from previous iterations  as long as the convexifications remain valid within tolerance. Another possible motivation is that noise arising from reusing previous convexifications may help avoid infeasible stationary points, as happens with SGD.
V Results
Va Experimental setup and implementation
Performance of each optimization method varies greatly with the choice of initialization, especially in the presence of nonconvex constraints such as obstacle avoidance. Optimizer behavior close and far from collision may also vary greatly, and therefore be informative of the advantages and disadvantages of each method. Because of this, in this paper we obtain results for two different initializations:

“Good” initialization: the robot’s base is placed such that its feet are above their target pose (hence not in collision with the ground). In particular, the optimization will start from such that the feet soles’ height will be equal to the maximum target foot sole height plus a margin of 0.20 meters. The joints are always set to a nominal posture (knees bent 50 degrees), and the base’s XY position is set to the center of the active contacts.

Initialization “incollision”: the robot’s base is placed such that its feet are in collision with the ground. In particular, the optimization will start from such that the feet soles’ height will be equal to the minimum target foot sole height minus a margin of 0.05 meters. The joints are always set to a nominal posture (knees bent 50 degrees), and the base’s XY position is set to the center of the active contacts.
As is common in optimization methods for robotics, we use “multistarts” (a.k.a. restarts) to deal with optimization failure. This means that in case the optimization does not converge from a certain initialization, we restart it from scratch using a slightly perturbed initial condition. In our experiments we restart the optimization up to 10 times, and each time perturb all joint angles using a uniform distribution in the interval of
degrees. We also report results without using multistarts, although they have expectedly lower success rates.We solved all problems with full, stochastic and incremental gradients, with sample sizes of . On average we obtained bestperforming results using for stochastic methods, for incremental gradient methods and for ISQP. For brevity we report results using these sample sizes only. Method SQP with is equivalent to ISQP but without memory (i.e. not incremental). Although we do not expect the algorithm to score high, because each QP will be very different from the next as certain constraints are removed, we still include it in the comparison for thoroughness.
We used cost and constraint implementations directly from the TrajOpt library [1]
. Although it is not part of the trajectory optimization task’s requirements, we also included a continuous collision constraint between waypoints to obtain trajectories that are closer to interpolationfeasible.
Due to the large number of variables on the trajectory optimization task, we solve each trajectory optimization problem as a sequence of optimization problems which consider only 6 stances (12 waypoints), with an overlap of 1 stance that is replanned. The first of those waypoints is fixed (planned in the previous window).
We implemented all optimization methods in C++ as extensions to the TrajOpt library. The same implementation for the costs, constraints and respective gradient computations is used for all optimization methods  thus making success rates and computation times directly comparable. The only exception is the implementation of penalty function gradients, which in the case of SQP and ISQP uses slack variables, while other methods use (16).
We compute the stepsize of each step in gradientbased methods through linesearch, using a zoom function and Wolfe’s conditions (, ), as in [21]. For Adam and Nadam we use standard parameters and . The termination criteria is as in [1]
for SQP methods, and is based on proportionaldecrease monitoring for gradient descent methods (i.e. terminate after 10 epochs without improvement in the objective function). Problems are solved successively with penalty coefficients
. We use Bullet for meshcollision, and solve QPs and feasiblespace projections using Gurobi.VB Algorithm comparison
We show all results for both tasks condensed in Table I.
Posture generation  Trajectory optimization  
Goodinit  Initincollision  Goodinit  Initincollision  
Method 
Multi  Sample  Success  Cost  Time  Success  Cost  Time  Success  Cost  Time  Success  Cost  Time 
starts  size  rate  (s)  rate  (s)  rate  (s)  rate  (s)  
SQP 
0  100  49/50  427  1.32  43/50  416  1.43  19/50  0.13  0.63  11/50  0.12  0.67 
SGD  0  100  44/50  720  3.76  41/50  770  3.55  4 /50  0.20  1.38  2 /50  0.18  1.26 
Adam  0  100  42/50  606  4.15  46/50  616  3.78  18/50  0.14  1.22  2 /50  0.14  1.27 
Nadam  0  100  39/50  590  2.83  48/50  645  2.33  16/50  0.14  0.81  1 /50  0.11  0.92 
SQP  10  100  50/50  427  1.44  50/50  429  1.74  42/50  0.13  0.62  29/50  0.12  0.84 
SGD  10  100  49/50  798  6.30  48/50  783  5.11  27/50  0.17  2.17  29/50  0.16  3.04 
Adam  10  100  49/50  643  5.97  50/50  672  4.30  31/50  0.14  1.50  32/50  0.15  2.10 
Nadam  10  100  49/50  642  5.67  50/50  679  3.20  35/50  0.15  1.19  30/50  0.15  1.96 
SQP  10  80  16/50  298  2.74  20/50  279  4.00  0 /50  0.00  0.00  0 /50  0.00  0.00 
SGD  10  80  50/50  788  5.82  49/50  775  4.44  26/50  0.16  1.99  30/50  0.16  2.49 
Adam  10  80  50/50  629  4.18  50/50  642  3.91  35/50  0.14  2.05  36/50  0.15  2.12 
Nadam  10  80  50/50  620  2.90  50/50  638  2.70  33/50  0.14  1.10  34/50  0.14  1.47 
ISQP  10  80  50/50  461  1.00  50/50  468  1.34  42/50  0.13  0.53  38/50  0.14  0.71 
ISGD  10  40  49/50  833  4.13  49/50  810  5.17  30/50  0.15  1.30  30/50  0.15  1.40 
IAdam  10  40  50/50  661  3.27  50/50  668  3.66  40/50  0.14  1.07  34/50  0.15  1.34 
INadam  10  40  50/50  667  2.66  50/50  684  3.44  41/50  0.15  0.73  28/50  0.14  1.64 

Note: Costs and times are averaged over successfully solved problems.
As expected, the use of multistarts greatly improves performance on all methods and conditions. In general, success rates are higher for the posture generation than the trajectory optimization task. This is understandable because trajectory problems consist of hundreds of postures (see Figure 3), while posture generation problems consist of only one. Success is also higher for the “goodinitialization” condition compared to the “incollision” condition, which proves the difficulty of optimization methods in dealing with collision constraints. Most of the failures in this condition consist of the optimization method getting stuck at an infeasible stationary point where no progress is made in any degree of freedom (i.e. collision’s signed distance gradient is inconsistent between links or cancels out the gradient of other constraints). We show an example of such a case in Figure 5.
Stochastic and incremental gradientdescent methods, as well as our ISQP, consistently lead to higher success rates than their fullgradient counterparts. SQP and ISQP have the lowest computation times, of 1 second per waypoint on average. In fact, ISQP is 20 to 30% faster than SQP on average. Gradientdescent variants are slower. IAdam takes 1 to 3 seconds per waypoint on average, while the Nesterovaccelerated INadam is around 15% faster on average. When it succeeds in finding a solution, SQP consistently obtains the lowestcost postures and trajectories, which we assume to be due to the nonexistence of noise as well as an efficient use of second order information and conditioning employed by the QP solver (Gurobi). The best performing method in both success rate and computation speed is always ISQP. Its cost values are similar but slightly higher than SQP.
For the posture generation task, on average all 50 problems are successfully solved by all methods. Computation speed is understandably faster for SQP and ISQP. This is because they use secondorder information and only recompute gradients after convergence of each QP, while gradient methods recompute gradients after each step (i.e. after each line search). In their stochastic and incremental versions, both SGD and SQPbased methods decrease computation time on average, some methods obtaining up to 50% speedups (IAdam).
The most common reason for algorithm failure is due to collision constraints, an example of which is shown in Figure 6. In this situation, SQP fails and requires several random restarts until it finds a feasible solution. ISQP and all stochastic and incremental gradient methods succeed at first try. The reason is that SQP gets stuck on an infeasible local minimum when a foot is in collision, while noisy gradients in the incremental/stochastic methods help move the robot out of collision (i.e. out of the local minimum).
For the trajectory optimization task, ISQP is the best performing in both the wellinitialized and incollision conditions, solving 42 and 38 problems respectively. ISQP considerably improves the success rate with respect to traditional SQP when initialized incollision (38 vs 29 successes). IAdam performs similarly, although slightly worse than SQP on wellinitialized problems (40 vs 42). However, IAdam is better than traditional SQP when started incollision (34 vs 29). In fact, all stochastic and incremental gradientbased methods outperform traditional SQP incollision, except for INadam.
Figure 4 shows a sequence of waypoints from a trajectory optimization problem where SQP fails, but ISQP succeeds, and Adam (samplesize ) also succeeds. SQP gets stuck on a local minimum where the left arm is in collision with a wall. However, ISQP successfully finds a posture which brings the arm out of collision, and Adam does so with a simpler posture closer to its nominal value (i.e. straight body with bent knees).
Vi Conclusions
In this paper we introduced a new challenge  LegOpt  for benchmarking different algorithms at the tasks of legged robot posture generation and trajectory optimization. LegOpt is open to public and defines a robot model, scenario, constraints and cost functions. Evaluation scripts are also made public for ease of use. LegOpt will hopefully help the legged and humanoid robotics communities to better compare the advantages and disadvantages of planning algorithms on common problems.
We evaluated not only stateoftheart algorithms based on sequential convex optimization (SQPs), but also recent stochastic and incremental algorithms introduced for largescale optimization in general and deep learning in particular. The assumption was that their robustness to local minima and saddle points would transfer to the optimization landscapes of robot optimization tasks as well. The experimental results showed that stochastic and incremental methods are indeed effective in the LegOpt Benchmark, especially on the presence of complex obstacle geometries or on initializations that are in collision. In particular, stochastic and incremental variants of SGD and Adam surpassed SQP when initialized incollision. Inspired by these algorithms we proposed an incremental version of SQP for optimizing robot motion, which updates only a random subset of the costs and constraints at each iteration. The algorithm obtained the highest success rates on tough initializations, and was consistently faster than traditional SQP.
Traditional and incremental SQP each performed better in different criteria. SQP arrived at better cost values when it found a solution, while ISQP was better regarding success rates by avoiding infeasible local minima. We believe results can be improved further by the use of continuation methods or more elaborate stochastic methods.
We realize many algorithms have been left out of this benchmark, of notice for example CHOMP and its variants. We hope the community will step forward and report the performance of different algorithms on LegOpt. We will also pursue the evaluation of new methods, with a special interest in stochastic methods. Although in this paper we define tasks and benchmark algorithms for a biped robot on the easiestdifficulty scenario from DSD [16], we envision the extension of the benchmark to multicontact tasks and tougher DSD scenarios. An extension to locomotion with dynamics is also important future work. The pace of these extensions will depend on progress made in this current version of the benchmark.
Acknowledgment
We would like to thank for developing and opening the Destruction Scenarios Dataset, as well as helping us adapt it to our needs.
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