Optimal Control Approach for Non-prehensile Ball Juggling Using a 7-DoF Manipulator

arXiv:2606.06704 · cs.RO · Submitted 2026-06-04 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: We're looking at the paper: "Optimal Control Approach for Non-prehensile Ball Juggling Using a 7-DoF Manipulator".

Dev: Here's the gist of the paper. The gist: A model-based framework for generating juggling trajectories and stabilizing a periodic juggling motion for this hybrid system is presented, incorporating a two-stage optimal control approach to compute the underlying feasible motion patterns required for stable juggling<ref:2606.06704#pg2> How it works The framework involves a swing-up phase followed by controlled juggles, which can be followed through Fig.

Rosa: 3<ref:2606.06704#pg4>.

Dev: The process begins by detailing the modelling and hybrid dynamics definition for the tool-ball interaction in Section III-A<ref:2606.06704#pg6>.

Rosa: The task is divided into two main stages, each formulated as a distinct OCP.

Paper summary: Taro: So we're wrapping up this look at "Optimal Control Approach for Non-prehensile Ball Juggling Using a seven-DoF Manipulator" <ref:2606.06704#pg1,Optimal Control Approach for Non-prehensile Ball Juggling Using a 7-DoF>. The authors are Joel Ramadani, Vasilije Rakcevi, Riddhiman Laha, Arne Sachtler, Valentin Le Mesle, Achim J. Lilienthal and Sami Haddadin.

Rosa: Yeah the title tells you exactly what we're looking at: optimal control for juggling with a seven degree-of-freedom manipulator. It's about getting a robot to juggle something without actually grasping it.

Dev: That tackles that core problem where juggling is hard because the object doesn't correct itself and its stability depends entirely on the forces you apply to it.

Taro: Exactly, and what they propose is this two stage model-based framework for generating those juggling trajectories and stabilizing the motion for this hybrid system.

Rosa: That framework uses a task-space optimal control problem to make a 2D reference trajectory for the tool first, and then a joint-space optimal control problem to map that down onto the full seven degree of freedom robot <ref:2606.06704#pg2>.

Taro: And they figure out how to manage that hybrid nature by alternating between contact and free fall using something called mathematical programs with complementarity constraints.

Rosa: They also use this framework offline to generate a database of pre-computed correction trajectories for when the ball's apex state gets slightly perturbed from the ideal one.

Dev: That means when they actually run it, instead of solving a huge optimal control problem in real time, they just look up the closest trajectory in that database based on where the ball is right now.

Taro: And to make sure everything lines up perfectly with timing, they use a mechanism to rescale the tool's trajectory timing continuously so there's no lag or asynchrony between releasing and catching.

Rosa: It looks like the authors think this is a way to get a continuous juggling pattern by using these offline calculations and then just applying a simple decision policy online.

Dev: It does sound like they are solving the problem of achieving dynamic stabilization in these underactuated systems without needing continuous, real-time optimal control solves for every tiny deviation.

Taro: The experimental validation showed they could sustain a certain number of juggles before the ball was dropped and that stability map across different contact parameters.

Rosa: It means for someone who only listens to the show, this paper suggests you can design mathematically stable juggling patterns and then use a simpler decision policy to correct errors based on where the ball actually is.

Dev: The implication for real-time control is that instead of throwing a massive computation at it during flight, you pre-calculate those correction trajectories for predicted states and just pick the best one.

Taro: It also shows how you can use trajectory re-parametrization to prevent timing issues caused by deviations between the tool release and ball apex.

Rosa: It's about having a structured way to generate stable juggling patterns that are then corrected by looking up pre-computed trajectories based on the observed ball state.

Dev: So, for this paper, the main thing is that you can break down a very dynamic, coupled system into manageable subproblems—a 2D task space problem and a full robot dynamics problem <ref:2606.06704#pg2>.

Taro: It's about creating a structured way to achieve dynamic stabilization in these non-prehensile manipulation tasks.

Conclusion: Rosa: So to wrap up this look at "Optimal Control Approach for Non-prehensile Ball Juggling Using a seven-DoF Manipulator," we've seen how they use a two stage model-based framework to generate and stabilize juggling trajectories for that complex seven degree of freedom robot.

Dev: Yeah, the title itself tells you what this paper is about: finding the optimal control method for juggling something without grasping it using a seven degree of freedom manipulator. It’s about getting that robot to juggle without actually taking hold of the ball.

Taro: What they're really tackling there is that core problem where juggling is hard because the object doesn't correct itself, and its stability depends entirely on the forces you apply to it, which makes timing absolutely critical for control inputs.

Rosa: And what this paper proposes is this systematic method for controlling a seven degree of freedom manipulator performing non-prehensile ball juggling with a tool. Their main contribution is that model-based framework for generating those juggling trajectories and stabilizing the motion for this hybrid system.

Dev: That framework uses a two stage optimal control approach to compute the underlying feasible motion patterns required for stable juggling, which they organize offline to enable real-time error correction without solving optimal control problems online.

Taro: It's about creating a structured way to achieve dynamic stabilization in these non-prehensile manipulation tasks by breaking down the dynamics into separate problems in different spaces.

Rosa: It's a lot of math to get there, but it gives us a robust way to stabilize objects without needing continuous online optimization for every single tiny deviation.

Dev: The limitation they flag is that because they have to solve the underlying optimal control problem, it exceeds the millisecond-scale time budget imposed by the task dynamics, which means real-time trajectory adaptation via online optimization becomes impractical.

Taro: That's a fair point about real-time computation constraints.

Rosa: So, for someone just listening to this show, what does this paper mean? It means we can design juggling patterns that are mathematically stable and then use a simpler decision policy to correct errors based on where the ball actually is.

Dev: It shifts the focus from solving a massive online optimization problem during the actual juggling motion to pre-computing solutions for common error states and just picking the best one.

Taro: It also shows how you can use trajectory re-parametrization to prevent timing issues caused by deviations between the tool release and ball apex.

Rosa: It’s about having a structured way to generate stable juggling patterns that are then corrected by looking up pre-computed trajectories based on the observed ball state.

Dev: This work shows how to generate juggling motions from an analytical two dimensional model and map these onto dynamically feasible joint space trajectories, which is a key part of this paper.

Taro: The authors are Joel Ramadani, Vasilije Rakcevi, Riddhiman Laha, Arne Sachtler, Valentin Le Mesle, Achim J. Lilienthal and Sami Haddadin.

Rosa: They did this work with the goal of developing a systematic method for controlling a seven degree of freedom manipulator performing non-prehensile ball juggling with a tool.

Dev: This paper is really about the framework they used to generate juggling trajectories and stabilize a periodic juggling motion for this hybrid system.

Taro: It addresses the problem that stability of an underactuated object depends entirely on the forces applied to it, making timing critical to continuously counteract deviations and maintain desired behavior.

Rosa: The implication is that for high speed manipulation tasks with sensitive dynamics, you can use this framework to get stable results without needing online optimal control solving during execution.

Dev: So, if you're in the field of real-time control, this suggests a way to pre-calculate correction trajectories for predicted perturbed apex states and deploy an online decision policy.

Taro: It also shows how you can break down a very dynamic, coupled system into manageable subproblems—a two dimensional task space problem and a full robot dynamics problem.

Rosa: It's about creating a structured way to achieve dynamic stabilization in these non-prehensile manipulation tasks.

Dev: This paper demonstrates effectiveness in both simulation and hardware using a Franka Panda robot, showing stability based on the total number of consecutive juggles sustained.

Taro: The authors are Joel Ramadani, Vasilije Rakcevi, Riddhiman Laha, Arne Sachtler, Valentin Le Mesle, Achim J. Lilienthal and Sami Haddadin.

Rosa: They did this work with the goal of developing a systematic method for controlling a seven degree of freedom manipulator performing non-prehensile ball juggling with a tool.

Dev: This paper is really about the framework they used to generate juggling trajectories and stabilize a periodic juggling motion for this hybrid system.

Dev: So, if you're in the field of real-time control, this suggests a way to pre-calculate correction trajectories for predicted perturbed apex states and deploy an online decision policy.

Taro: It also shows how you can use trajectory re-parametrization to prevent timing issues caused by deviations between the tool release and ball apex.

Joel Ramadani, *, Vasilije Rakcevic, Riddhiman Laha, Arne Sachtler, Valentin Le Mesle, Achim J. Lilienthal

Technical University of Munich Department of Robotics and Mechatronics Institute for Robotics and Mechatronics German Aerospace Center

cs.RO

Submitted: 2026-06-04

Updated: 2026-06-04

Comments: 8 pages, accepted at ICRA 2026

Journal ref: 2026 IEEE International Conference on Robotics and Automation (ICRA), Vienna, Austria, 2026, pp. 5805-5812

DOI: 10.1109/ICRA57385.2026.11696404

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 75/100

The gist: The gist: A model-based framework for generating juggling trajectories and stabilizing a periodic juggling motion for this hybrid system is presented, incorporating a two-stage optimal control

Key concepts

Task-Space OCP: This is the first optimal control problem that designs the desired motion path for the tool in a 2D plane. It aims to generate an ideal trajectory based on a simple reference model, accounting for the hybrid nature of contact and free fall using mathematical programming with complementarity constraints.
Joint-Space OCP: This second optimal control problem translates the desired 2D tool path into actual movements for the robot's 7 degrees of freedom. It ensures that the resulting joint trajectories respect physical limits like joint limits, velocity constraints, and torque limitations while tracking the task-space path.
Lagrange Multiplier ($\lambda$): In this dynamic model, the Lagrange multiplier $\lambda$ is used to enforce geometric constraints on the ball's interaction with the tool. Physically, it represents a scaled normal force exerted by the tool onto the ball, allowing for a more accurate description of how forces influence stability.
Trajectory Re-parameterization: To maintain timing accuracy during juggling, an adaptive mechanism rescales the calculated trajectory timing. This is done by solving a boundary value problem to create a new smooth timing law, ensuring that the tool's motion transitions between phases remains continuous and synchronized.

Terminology

Hybrid System — 1× in this paper · explained in 2 episodes

Summary

The gist: A model-based framework for generating juggling trajectories and stabilizing a periodic juggling motion for this hybrid system is presented, incorporating a two-stage optimal control approach to compute the underlying feasible motion patterns required for stable juggling<ref:2606.06704#pg2>

How it works

The framework involves a swing-up phase followed by controlled juggles, which can be followed through Fig. 3<ref:2606.06704#pg4>. The process begins by detailing the modelling and hybrid dynamics definition for the tool-ball interaction in Section III-A<ref:2606.06704#pg6>. The task is divided into two main stages, each formulated as a distinct OCP.

Stage 1. Task-Space OCP – generates an ideal reference trajectory for the tool’s motion based on a 2D reference model.
Stage 2. Joint-Space OCP – takes the 2D tool trajectory and maps it onto the full robot model.

Dynamics Model

The dynamics of the ball within a freely movable tool are described using Lagrangian formalism extended by a Lagrange multiplier λ to enforce geometric constraints. The specific Lagrangian for the system is given by L = 1/2 m x˙2b + y˙2b - mg yb z T - mgyb z U + λ (xb −xc)2 + (yb −yc)2 - r2 C(3). The Lagrange multiplier λ can be physically interpreted as the normal force scaled by the tool’s diameter, i.e., λ = N/2r.

Task-Space Optimal Control for Tool Trajectory Generation

The Task-Space OCP is formulated to generate tool motions where the control input u(t) is the tool’s Cartesian acceleration, u(t) = [x¨c(t),y¨c(t)]T. A key challenge addressed is the system’s hybrid nature, alternating between contact and free fall, which is managed by formulating the OCP as a Mathematical Program with Complementa Constraints (MPCC). The complementarity condition is parametrized by a smooth signed distance function φ(x,α), representing log-sum-exp. The complete OCP minimizes the control effort J = integral Z tf 0 u(t) squared dt subject to the system dynamics and boundary conditions.

Joint-Space OCP for Feasible Reference Trajectory Tracking

To translate the generated 2D task-space trajectories into physically feasible joint trajectories for the 7-DoF robot, a second OCP (12) is defined. This Joint-Space OCP finds a feasible jointspace trajectory that follows the desired tool trajectories while simultaneously maintaining the tool in a desired orientation. Constraints include joint limits qmin ≤ q(t) ≤ qmax (12b), velocity limits ur(t) ≤ 0.9 ur,max (12c), and torque limits ur (12c). The dynamics constraint encodes the standard robot equations of motion: x˙r = q˙ M(q)−1 ur − C(q,q˙)q˙ − g(q).

Stabilization of the Juggling Motion

Since optimized trajectories do not exhibit self-stabilizing properties under open-loop execution, a feedback mechanism adapts the tool’s motion once per cycle based on the observed ball state. For deciding on a trajectory for the next cycle, a ball state at release is measured, and its predicted apex state xˆb is matched to the closest entry in the pre-computed trajectory database using minimum Euclidean distance dmin. The database contains trajectories associated with all perturbed apex states of the ball (a square grid of apex positions spanning 5 cm × 5 cm around the desired apex, with a resolution of 3.125 mm).

Trajectory Transitioning via Time Reparameterization

To prevent asynchrony caused by timing deviations, an adaptive mechanism is implemented to rescale the tool’s trajectory timing. This is achieved by computing a new, smooth timing law snew(t) as the solution to a boundary value problem for the interval [trelease, tapex]. The desired joint position vector is then synthesized from this new timing law:⃗qdes(t) =⃗qpath(snew(t)). This ensures C1-continuity at the transition point trelease.

Experimental Validation

Performance was evaluated in both simulation and hardware experiments using the Franka Panda robot with a circular tool and a ball to perform swing-up and juggling tasks. The experimental friction coefficient of µ = 0.105 is used for trajectory generation. The controller’s sensitivity was analyzed via a parameter sweep, showing stability measured by the total number of consecutive juggles (apex count) the system could sustain before the ball was dropped. The results were visualized in Fig. 8a, which shows the stability map of the control approach across different contact parameters. The hardware experiments are limited to a swing-up and single juggle cycle for evaluating the presented model-based framework and robot performance abilities.

Conclusion

A model-based two-stage optimal control framework is presented, where the Task-Space OCP generates juggling motions from an analytical 2D model, and the Joint-Space OCP maps these to dynamically feasible jointspace trajectories. This process is repeatedly solved offline to generate a library of pre-computed correction trajectories for the predicted perturbed apex state range. A decision policy is then deployed online to select the next correction trajectory based on the measured ball state. This approach ensures a continuous and stable juggling pattern through trajectory re-parametrisation.

Acknowledgement

The authors would like to thank Bingkun Huang for fruitful discussions and help with the hardware experiments. The work has partially been supported by the German Federal Ministry of Research, Technology and Space (BMFTR) under the Robotics Institute Germany (RIG), grant no. 16ME0997K.

References

[1] F. Ruggiero, V. Lippiello, and B. Siciliano, “Nonprehensile dynamic manipulation: A survey,” IEEE Robotics and Automation Letters, vol. 3, no. 3, pp. 1711–1718, 2018.

[2] R. I. C. Muchacho, R. Laha, L. F. Figueredo, and S. Haddadin, “A solution to slosh-free robot trajectory optimization,” in 2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[3] M. Buehler and D. E. Koditschek, “Robotics in an intermittent dynamical environment: A prelude to juggling,” 1987.

[4] E. W. Aboaf, C. G. Atkeson, and D. J. Reinkensmeyer, “Task-level robot learning,” in Proceedings of 1988 IEEE International Conference on Robotics and Automation.

[5] S. Schaal, C. G. Atkeson, and D. Sternad, “One-handed juggling: A dynamical approach to a rhythmic movement task,” Journal of Motor Behavior, vol. 28, no. 2, pp. 165–183, 1996<ref:

Improvements for AI systems

Improved AI system can execute high-speed, non-prehensile manipulation tasks for robots by using a model-based two-stage optimal control framework to generate feasible juggling trajectories. This system can perform complex maneuvers like swing up and periodic juggling of balls with a tool while respecting physical constraints through a Joint Space OCP.
Improved AI system can achieve real-time error correction during juggling by employing an online decision policy that selects the most suitable recovery trajectory from a pre-computed database based on the measured ball state, as described in Section IV A.
Improved AI system can maintain continuous juggling behavior even when executing periodic motions by using a trajectory re-parametrization approach to ensure C1 continuity between cycles, preventing timing deviations during online execution.

Listen

Transcript

Key concepts

Terminology

Summary

Improvements for AI systems

Related papers