I Introduction
Fast and reliable robot pickandplace motion is increasingly a bottleneck on modern automated warehouses. Motions that are too slow, while usually safe, reduce robot throughput, while motions that are too fast can be unreliable, leading to dropped objects due to shearing forces between the object and the gripper that holds it. Our prior work, GraspOptimized Motion Planning (GOMP [1]), showed that simultaneously optimizing grasp pose and pickandplace motion could allow for rapid object transport. However, given the torque available to modern industrial robot arms, the highspeed motions that GOMP generates could lead to lost grasps, product damage, and spills. In this work, we propose a fast inertial transport (FIT) problem and algorithm to address it, in which a robot transports objects at high speeds, banking and limiting motions against inertial forces to allow the robot to safely and reliably place the object at its target location.
To enable FIT, we propose GraspOptimized Motion Planning for Fast Inertial Transport (GOMPFIT) which computes timeoptimized motions while taking into account endeffector and object acceleration constraints imposed by the object being transported and how it is grasped. GOMPFIT incorporates constraints for opentop container transport, fragile object transport, combinations thereof, and potentially more. For opentop containers, GOMPFIT aligns inertial accelerations so that the contents remain in the container. For fragileobject transport, GOMPFIT limits the magnitude of accelerations of the transported object to avoid exceeding a shock threshold that could damage the object. For fragile opentop containers, GOMPFIT constrains both magnitude and alignment of accelerations. As most industrial robot arms do not provide direct access to motor torques, GOMPFIT instead operates in the joint configuration space, relying on the robot arm’s blackbox controller to follow trajectories.
In experiments, we task a physical UR5 robot with transporting various objects with endeffector acceleration requirements, including: transporting cups filled to various levels with beads without spilling their contents, transporting an IMU to measure accelerations that a fragile object would experience in transit, and moving a filled wineglass. Experiments suggest that GOMPFIT can achieve near 100 % success on these tasks while slowing as little as 0 % when there are few obstacles, 30 % when there are high obstacles and 45degree tolerances, and 50 % when there are 15degree tolerances and few obstacles compared to GOMP. This work contributes:

a formulation of the fast inertial transport motion planning problem,

a sequential convex program formulation, including nonconvex constraints on accelerations at the endeffector, and a sequential quadratic program implementation to solve it,

data from experiments on a physical UR5 robot transporting cups filled with beads, a fragile object and a filled wineglass.
Ii Related Work
Iia Motion Planning and Optimization
Motion planning seeks to produce paths from start to goal while avoiding obstacles. Samplingbased motion planners such as PRM [2], RRG [3], and RRT [4] have favorable properties such as probabilistic completeness and asymptotic optimality, but they may be too slow to converge.Such planners also typically require postprocessing optimization step to remove redundant or jerky motion. Optimizationbased motion planners such as TrajOpt [5], STOMP [6], and CHOMP [7] can produce smooth trajectories while fulfilling a set of constraints such as obstacle avoidance through interior point optimization, stochastic gradients, and covariant gradient descent. In prior formulations, the trajectory is discretized into a series of waypoints with a minimization objective such as sum of squares velocity. Motion planners such as GOMP [1] and DJGOMP [8] incorporate dynamics constraints and iteratively shorten horizon length to find a timeoptimized trajectory, and apply jerk limits to avoid damaging the robot. Unlike previous work, this paper applies additional constraints to the end effector acceleration to prevent an object held by a robot arm from damage or detachment in highacceleration trajectories, something that constraints on configurations and their derivatives alone cannot do.
IiB EndEffector Constraints
Placing constraints on the endeffector motion path is a requisite part of many problems, including fast inertial transport. Yao and Gupta [9] address the pathplanning problem with general endeffector constraints by exploring the task space for feasible endeffector poses through a samplingbased planner. Li et al. [10] propose using RL to compute motion plans for dualarm manipulators with floating base in space. They propose to take into account a velocity constraint of the endeffector in the planning process to improve the robustness of the algorithm. But these methods do not support acceleration constraints for the fast inertial transport problem we propose.
IiC Planning for Dynamic Manipulation
Dynamic manipulation research uses dynamic properties such as momentum to achieve a task—whether through prehensile or nonprehensile manipulation. Lynch and Mason [11]
exploit centrifugal and Coriolis forces to manipulate objects using lowdegreeoffreedom robots. Lynch and Mason
[12] also formulate an SQP for dynamic nonprehensile manipulation of objects, such as snatching, throwing, and rolling. They directly integrate constraints based on a 2D formulation of the problem but do not consider obstacles. Srinivasa et al. [13] propose employing constraints on accelerations at the end point to perform dynamic object manipulation—specifically nonprehensile rotation of an object. Kim et al. [14] propose a method to rapidly computes motions to catch an object in flight. Mucchiani and Yim [15] propose a method to use inertial effects to dynamically sweep up an object and stabilize it using a passive endeffector. In contrast, we propose a 3D prehensile planner for safe and reliable object transport around obstacles.A promising line of research, especially for objects with unknown or difficulttomodel dynamics, is to employ learning for dynamic manipulation. Zeng et al. [16] propose TossingBot that learns parameters of a predefined dynamic motion to throw objects into target bins. Zhang et al. [17] propose Robots of the Lost Arc that manipulates a rope through a sweeping dynamic motion to hit targets, weave through obstacles, or knock objects down. Wang et al. [18] propose SwingBot, that uses inhand tactile feedback to learn how to dynamically swing up previously unseen objects. These methods rely on a learned model for dynamic manipulation, while we propose using Newtonian physics.
Another promising approach is to integrate dynamics considerations into samplingbased planning. Pham et al. [19, 20] proposed admissible velocity propagation (AVP) and a method to perform kinodynamic planning in a reduced dimensionality state space, and Lertkultanon and Pham [21] formulate a ZMP constraint for AVP and integrate bidirectional RRT to nonprehensile object transportation. Unlike the AVP formulation, GOMPFIT integrates all constraints, including collision, in a single optimization.
To enable realworld interaction between robots and the environments they touch, researchers have looked into optimizing motions in the presence of contacts. Posa and Tedrake [22] and Posa et al. [23] propose simultaneously optimizing trajectories and contact points. Hauser [24] proposes a trajectory optimization that considers contact forces and convex time scaling. Luo and Hauser [25] propose integrating feedback and learning to integrate confidence into the optimization, and apply it to the Waiter’s Problem of stably transporting objects on a moving platform. While these methods consider contacts, they do not consider obstacle avoidance.
IiD Dynamic Object Transport
Most closely related to this paper is research into transport of objects that takes into account inertial effects. Bernheisel and Lynch [26] propose a Waiter’s Problem in which object assemblies are stably transported subject to inertial and gravity forces. They focus on multipart assemblies and assume quasistatic motion in which only the velocity direction matters. In contrast we focus on a containment and accelerationbased inertial effects. Wan et al. [27] demonstrates beverage transport using jerkconstrained trajectories for a mobile base, showing that jerklimits help prevent spills during transport. In contrast, we show that jerklimits alone will not prevent spills when using a manipulator arms. Acharya et al. [28] propose methods to compute minimumtime scurve trajectories and apply it to the Waiter’s Problem. To transport objects with unknown mass parameters, Lee and Kim [29]
perform online estimation of unknown parameters of the payload to integrate into the trajectory generation for cooperative aerial manipulators. For fast transport of objects held in a suction grasp, Pham and Pham
[30] propose combining RRT with a trajectory time parameterizer that remains within the suction contact stability constraint. In contrast, we propose a single optimization process that incorporates all necessary constraints to perform fast inertial transport.Iii Problem Statement
Let be the complete specification of a robot’s degrees of freedom, where is the space of all configurations. Let be the set of configurations that are in an obstacle, and be the set of configurations that are not in collision. Let be the forward kinematics function that computes the pose of the object in the robot’s endeffector. Let be an inverse kinematic (IK) function that computes the robot’s configuration, given a desired location of the endeffector. The IK function may not always have a solution, and may not have a unique solution. Let
be the dynamic state of the robot at any moment in time, including the first and second derivative of the robot’s configuration. Let
be the linear acceleration at the endeffector including gravity and the inertial (fictitious) forces: Euler, Coriolis, and centrifugal.Given a starting grasp and a placement pose , and constraints on the endeffector accelerations, the objective of GOMPFIT is to compute a trajectory , where is the duration, , , , and the endeffector acceleration constraints are met at all . Furthermore, the objective is to minimize the total trajectory time, subject to the robot’s actuation limits. Thus,
where is the trajectory’s duration in seconds (and referenced without parameters equivalently), and is the set of problemspecific endeffector acceleration constraints, such as keeping opentop container contents or avoiding excessive shock.
Iv Method
To compute a highspeed motion that remains within endeffector acceleration constraints, we first discretize the problem, then formulate a nonconvex optimization, and finally solve the optimization using a sequence of sequential quadratic programs based on TrajOpt [5] and GOMP [1].
Iva GOMP Background
GOMPFIT is built on GraspOptimized Motion Planning (GOMP). GOMP computes a timeoptimized obstacleavoiding trajectory that incorporates a degree of freedom around pick and place points. This section reviews the GOMP formulation and optimization process.
IvA1 Trajectory Discretization
To facilitate solving the GOMPFIT optimization problem, we first formulate a discretization of the trajectory. The discretization serves a second purpose—all industrial robots operate with a fixed control frequency, and can follow trajectories at only that rate or an integer multiple of it.
GOMP and GOMPFIT first define a fixed number of waypoints , each separated by a fixed time step . The time step should be an integer multiple of the robot’s control frequency, and should be sufficient to allow the problem to be solved (more details in Sec. IVA4). Each waypoint includes a configuration and first and second derivatives and acceleration .
To approximate and , we have the optimization process enforce a dynamics constraint between each waypoint in the follow form:
(1)  
(2) 
This discretization comes from integrating a linear jerk between waypoints.
IvA2 Sequential Convex Optimization
To solve the discretized problem, we formulate and solve a sequential quadratic program of the following form:
s.t. 
where is a positive semidefinite matrix for the quadratic costs,
is a linear cost vector, and the matrix
and vector define the linear constraints.We construct and to include the convex constraints: (a) the dynamics constraints from equations 1 and 2; (b) the actuation limits of the robots, e.g., lower and upper bounds of configuration, velocity, and acceleration; and (c) jerk limits computed using the finite differences between waypoint accelerations.
To handle nonconvex constraints from (i) grasp and placement configurations and degrees of freedom; (ii) obstacle avoidance; and (iii) endeffector acceleration limits, we add linearizations of the constraints to and with slack variables constrained to be positive.
As with GOMP, we establish the trust region around the configurations of the waypoints, and leave the other optimization variables (e.g., and ) otherwise untouched.
IvA3 Linearization of NonConvex Constraints
To enable obstacle avoidance, graspoptimization, and now acceleration constraints, GOMP and GOMPFIT formulates a (nonconvex) constraint of the form , and linearizes the function around the current iterate as:
where is the Jacobian evaluated at , and is the optimization variable for the next iteration. These coefficients are then added to and .
IvA4 Time Optimization
To minimize trajectory time, GOMP and GOMPFIT repeatedly solve the optimization with a shrinking horizon until the SQP returns failure, and uses trajectory from the minimum horizon that succeeded.
IvB GOMPFIT EndEffector Acceleration Constraints
To place constraints motions base on inertial effects, we first compute the acceleration at the endpoint due to gravity, Euler, Coriolis, and centrifugal forces; and then use that in the formulation of constraints. To compute net effect of all joint motions on the endeffector acceleration, we employ the forward pass of the Recursive Newton Euler (RNE) method [31], summarized in Alg. 1. This computation requires the rotation between joints and ; the angular velocity vector and its derivative ; and the translation between successive joints . We do not employ the backward pass of the RNE, as we are only interested in computing the acceleration at the end effector.
With the RNE method, we compute the acceleration at the endeffector , where is the gravity constant vector. We also define as the forward kinematic to the grasp vector (Fig. 1)—i.e., normal of a container’s surface, or the suction contact normal. We then use this to formulate the nonconvex constraints needed for object transport. For notation convenience, we leave out the waypoint subscript , as these constraints apply to all waypoints individually.
IvB1 OpenTop Containers
For transporting opentop containers, we constrain trajectories to keep the combined acceleration at the endeffector within a userdefined threshold angle of the normal of the bottom surface of the container. This constraint has the form:
(3) 
This can equivalently be expressed as
The choice of constraint appears to have little effect on convergence.
IvB2 Fragile Objects
When transporting fragile objects, we constrain trajectories to avoid exceeding a threshold acceleration in any direction. This constraint has the form:
(4) 
IvB3 Combining Constraints
IvC Minimization Objective
As the constraints on endeffector accelerations and their linearizations depend on the velocity and acceleration of the configuration, we find that adding a minimization objective based on the sumofsquared velocities or accelerations can work against the linearizations. These objectives tend to “pull” trajectories against the constraints. We thus minimize the sumofsquared jerks of the trajectory using a finite difference between waypoints:
In practice, we find that assigning different weights to different joints benefits motion computation. Specifically, we weigh joints closer to the base higher than joints closer to the endeffector, encouraging those joints to focus more on producing a smooth overall motion with minimal jerk whereas joints near the gripper are encouraged to fulfill other problemspecific constraints such as the start goal constraint.
V Experiments
(a)  (b)  (c)  (d)  (e) 
We experiment with GOMPFIT on a UR5 physical robot performing a series of tasks in which success is dependent on constraints on the endeffector acceleration. These tasks are: (1) transporting an opentop container without spilling contents, (2) transporting a fragile object without exceeding an acceleration threshold (3) transporting wine held in a wine glass without spilling, or dropping the wineglass.
For baselines, we run both GOMP and JGOMP. This also forms an ablation in that GOMPFIT without the acceleration constraints is JGOMP, and JGOMP without the jerk limits and minimization is GOMP, as GOMP instead minimizes sumofsquared accelerations at the joints. Here, both GOMP and JGOMP include a minor modification so that all baselines share the same dynamics constraints as GOMPFIT. Additionally, we add baselines of GOMP(+H) and JGOMP(+H), which are GOMP and JGOMP but with the same minimum horizon as computed by GOMPFIT. These two baselines are to test whether it is just the slower speed of the trajectory, or if the additional constraints on endeffector accelerations are necessary for successful completion of the tasks.
Va OpenTop Container Transport
Tolerance  GOMP  JGOMP  GOMP(+H)  JGOMP(+H)  GOMPFIT 

45  89.7%  46.4%  84.5%  37.1%  0.0% 
30  90.0%  48.0%  47.0%  100.0%  0.0% 
20  90.3%  50.0%  39.4%  100.0%  0.0% 
15  91.2%  54.4%  41.2%  100.0%  0.0% 
Metric  GOMP  JGOMP  GOMPFIT  GOMPFIT 2G 

IE  330.9  121.7  73.6  82.5 
IVE  255.3  61.1  94.5  18.3 
To test if GOMPFIT can transport an opentop container without spilling the contents, we task a UR5 robot to transport glass containing beads (Fig. 3). We vary the fill level based on a tiltlevel. We tilt the container 45, 30, 20, and 15, and fill it so that the beads just barely stay and record the mass (Fig. 4). We then have GOMPFIT compute motions that include the opentop transport constraint (Eqn. 3) matching the fill level, and execute the motion. We vary this for 3 different startgoal pairs and report the results in Tab. I. From the results we observe that GOMPFIT does not drop a single bead, while all other planners do. With the (+H) ablations, we observe that running slower is not sufficient for task success, as both baselines spill contents at the same trajectory duration–some of this is attributable to the endeffector taking a longer path to match the trajectory length. Best performing of the baselines is JGOMP, which, while consistent with prior observations [27] that jerk limits reduce spills, limiting jerk alone is insufficient for fast inertial transport.
We also compare motion time to study how much time is lost due to the opentop constraint, and report the results in Fig. 5. The time loss is surprisingly small, as we observe that GOMPFIT is able to compute motions that tilt against the inertial forces to keep the contents in the container—most of the time lost is in accelerating and decelerating the object at the beginning and end of the motion. As obstacle height increases and angle tolerances tighten, the safe passage takes more time to traverse. With shorter obstacles, GOMP’s lack of jerk constraints, requires reducing accelerations limits to prevent protective stops, resulting in worse performance than JGOMP. Both JGOMP and GOMPFIT can run with higher acceleration limits.
VB Fragile Object Transport
Transporting a fragile object may require limiting the endeffector acceleration to avoid damaging or dropping the object. To test if GOMPFIT can reliably limit the endeffector acceleration, we attach a RealSense D435i camera with an Inertial Measurement Unit (IMU) to the UR5 robot endeffector and record the acceleration norm along the executed trajectories. We compute two metrics: the integrated error (IE) as the sum of absolute difference between the endeffector acceleration norm and the planned acceleration norm:
and the integrated violation error (IVE) as the difference between the endeffector acceleration norm and the acceleration limit:
In experiments we set = 2G = 19.74 m/s and compare GOMPFIT with a 2G acceleration limit, with the baselines GOMP, JGOMP, and GOMPFIT with a 45alignment constraint. The results are summarized in Tab. II and a qualitative result is presented in Fig. 6. From the measures, we observe that unlike baselines, GOMPFIT is able to accurately limit accelerations, subject to errors inherent to the sensor and underlying controller.
VC Filled Wineglass Transport
To test GOMPFIT for fast inertial transport of a liquid in an opentop container, we compute and execute a motion that transports a glass of wine over a barrier. Due to results of the experiments with the cup containing beads, and the damage (and stains) that would result from spills, we do not attempt any baseline method. This experiment is mostly qualitative, as we do not wish to probe the limits of the method in our current lab setup, but rest assured we did not spill or waste a drop. A preview of the accompanying video is in the topright of Fig. 1.
Vi Conclusion
We present GOMPFIT, an optimizing motion planner that solves the fast inertial transport problem of transporting opentop containers, fragile objects, or combinations thereof around obstacles while not spilling, damaging, or losing a grasp due to inertial forces. GOMPFIT uses a sequential quadratic program to optimize a discretized trajectory with the first and second derivatives of its configuration, and incorporates nonlinear constraints for obstacle avoidance, grasp optimization, and on accelerations at the endeffector. In experiments on a physical robot, GOMPFIT was able to reliably and safely transport objects with only minor slowdown compared to motions from fast planners that did not safely transport objects. Slowed motions of the baseline planners fared little better, suggesting that slowing motions down is not sufficient to achieve reliable inertial transport.
In future work, we will explore addressing the long computation time associated by tuning the optimizer and using deep learning to warm start the computation
[8]. The proposed method relies on conservative approximations for joint acceleration limits based on torque limits and transported mass—we hope to tighten up the acceleration limits by transitioning to torquebased limits. In the current formulation, these would be nonconvex constraints that would further slow the computation but result in faster motions.Acknowledgment
This research was performed at the AUTOLAB at UC Berkeley in affiliation with the Berkeley AI Research (BAIR) Lab, Berkeley Deep Drive (BDD), the RealTime Intelligent Secure Execution (RISE) Lab, and the CITRIS “People and Robots” (CPAR) Initiative. We thank our colleagues who provided helpful feedback and suggestions, in particular Matthew Tancik. This article solely reflects the opinions and conclusions of its authors and do not reflect the views of the sponsors or their associated entities.
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