Per-Contact Iteration Method for Solving Contact Dynamics
Open accessAbstract
This letter introduces a new iterative method for contact dynamics problems. The proposed method is based on an efficient bisection method which iterates over each contact. We compared our approach to two existing ones for the same model and found that it is about twice as fast as the existing ones. We also introduce four different robotic simulation experiments and compare the proposed method to the most common contact solver, the projected Gauss-Seidel (PGS) method. We show that, while both methods are very efficient in solving simple problems, the proposed method significantly outperforms the PGS method in more complicated contact scenarios. Simulating one time step of an 18-DOF quadruped robot with multiple contacts took less than 20 μs with a single core of a CPU. This is at least an order of magnitude faster than many other simulators which employ multiple relaxation methods to the major dynamic principles in order to boost their computational speed. The proposed simulation method is also stable at 50 Hz due to its strict adherence to the dynamical principles. Although the accuracy might be compromised at such a low update rate, this means that we can simulate an 18-DOF robot more than thousand times faster than the real time.
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Present-day technology is severely limited in its ability to simulate compliant contact, slippage, and deformable and crumbling terrain; phenomena such as mud, snow, thick vegetation, and gushing water are beyond the capabilities of robotics simulation frameworks.
“Present-day technology is severely limited in its ability to simulate compliant contact, slippage, and deformable and crumbling terrain. As a result, phenomena such as mud, snow, thick vegetation, gushing water, and many others are beyond the capabilities of robotics simulation frameworks [30-32].”
From Learning quadrupedal locomotion over challenging terrain (2020) · cites this paper as [30]
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BackgroundMethodCat-Like Jumping and Landing of Legged Robots in Low Gravity Using Deep Reinforcement Learning
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BackgroundMa, Y., Farshidian, F., Miki, T., Lee, J., & Hutter, M. (2022) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2022.3143567 · Cited by 87 · see citing works
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MethodSelf-citationLearning Free Gait Transition for Quadruped Robots Via Phase-Guided Controller
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Self-citationTrajectory Optimization With Implicit Hard Contacts
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BackgroundCircus ANYmal: A Quadruped Learning Dexterous Manipulation with Its Limbs
Shi, F., Homberger, T., Lee, J., Miki, T., Zhao, M., Farshidian, F., Okada, K., Inaba, M., & Hutter, M. (2021) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra48506.2021.9561926 · Cited by 55 · see citing works
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Hamed, K. A., Kim, J., & Pandala, A. (2020) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2020.3001471 · Cited by 54 · see citing works
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Fu, Z., Kumar, A., Agarwal, A., Qi, H., Malik, J., & Pathak, D. (2022) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr52688.2022.01676 · Cited by 51 · see citing works
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“We use the Raisim simulator 43 for training.”
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“We use the Proximal Policy Optimization (PPO) RL algorithm [30], chosen for its relative robustness towards hyperparameter choices, with fast CPU-parallelized simulation rollouts produced through the Raisim physics simulator [31].”
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Lidec, Q. L., Jallet, W., Montaut, L., Laptev, I., Schmid, C., & Carpentier, J. (2024) · IEEE Transactions on Robotics · doi:10.1109/tro.2024.3434208 · Cited by 45 · see citing works
“RaiSim [33] emerged as one of the first simulators enabling RL policies to transfer to real quadrupedal robots.”
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“The training of the MLP network is performed offline using numerical simulations with the learning environment provided by RaiSim [45].”
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“For low-level policy training, we used RaiSim [43] to simulate the robot and the environment, while for high-level policy, Isaac Gym [44] was used to simulate confined space environment with depth sensor measurements such as spherical scan, depth camera and voxel map.”
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Testing of Deep Reinforcement Learning Agents with Surrogate Models
Biagiola, M., & Tonella, P. (2023) · ACM Transactions on Software Engineering and Methodology · doi:10.1145/3631970 · Cited by 40 · see citing works
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BackgroundDifferentiable Dynamics for Articulated 3d Human Motion Reconstruction
Gärtner, E., Andriluka, M., Coumans, E., & Sminchisescu, C. (2022) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr52688.2022.01284 · Cited by 38 · see citing works
Zhang, W., Ott, L., Tognon, M., & Siegwart, R. (2022) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2022.3194315 · Cited by 38 · see citing works
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MethodComparing Popular Simulation Environments in the Scope of Robotics and Reinforcement LearningPreprint
Körber, M., Lange, J., Rediske, S., Steinmann, S., & Glück, R. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.04616 · Cited by 37 · see citing works
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Yang, B., Wellhausen, L., Miki, T., Liu, M., & Hutter, M. (2021) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra48506.2021.9561861 · Cited by 35 · see citing works
“1) Dataset: We simulate the ANYmal C in Raisim [16] with a state-of-the-art learning-based locomotion controller [17], which enables the robot to travel on challenging ter-”
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Chamorro, S., Klemm, V., Valls, M. D. L. I., Pal, C., & Siegwart, R. (2024) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra57147.2024.10610069 · Cited by 33 · see citing works
“Compared to other widely-used simulators such as Raisim [22] and MuJoCo [23], Isaac Gym’s GPU-accelerated architecture enables significantly faster training of agents, thanks to its high parallelization.”
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“Implementation Details We use PPO [54] for RL training and RaiSim [34] for the physics simulation.”
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“3) Implementation Details: The training environments are implemented using Raisim [26] simulator.”
Self-citationReal-Time Trajectory Adaptation for Quadrupedal Locomotion using Deep Reinforcement Learning
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“We use the RaiSim [34] physics simulator to train our RL policy.”
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Bi, T., Sferrazza, C., & Raffaello D’andrea, B. (2021) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2021.3084880 · Cited by 31 · see citing works
“Using a semi-implicit integration scheme ([28], [29]) the equations of motion are then given by”
Convex Optimization for Trajectory GenerationPreprint
Malyuta, D., Reynolds, T. P., Szmuk, M., Lew, T., Bonalli, R., Pavone, M., & Acikmese, B. A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2106.09125 · Cited by 31 · see citing works
Kim, J., Fawcett, R. T., Kamidi, V. R., Ames, A. D., & Hamed, K. A. (2023) · IEEE Transactions on Robotics · doi:10.1109/tro.2023.3319896 · Cited by 30 · see citing works
“2) Computation, Control Loop, and Network: We use RaiSim [92] to simulate both the interconnected reduced-and full-order models numerically.”
MethodKim, G., Kang, D., Kim, J.-H., Hong, S., & Park, H.-W. (2024) · The International Journal of Robotics Research · doi:10.1177/02783649241273645 · Cited by 29 · see citing works
“For 2) and 3) cases, RaiSim simulator (Hwangbo et al. 2018) with 1 ms integration time step is used, and the MPC runs at 40 Hz.”
InfluentialGym-Ignition: Reproducible Robotic Simulations for Reinforcement Learning
Ferigo, D., Traversaro, S., Giorgio Metta, B., & Pucci, D. (2020) · IEEE/SICE International Symposium on System Integration (SII) · doi:10.1109/sii46433.2020.9025951 · Cited by 29 · see citing works
“RaiSim [28] is a recently released simulator specific for robotics.”
BackgroundStructured learning of rigid‐body dynamics: A survey and unified view from a robotics perspective
Geist, A. R., & Trimpe, S. (2021) · GAMM-Mitteilungen · doi:10.1002/gamm.202100009 · Cited by 29 · see citing works
“[43] 2019 FD [44] NN Qu ✓ × Greydanus et al.”
MethodAn Unconstrained Convex Formulation of Compliant Contact
Castro, A., Permenter, F., & Han, X. (2022) · IEEE Transactions on Robotics · doi:10.1109/tro.2022.3209077 · Cited by 27 · see citing works
“RaiSim [23] implements an iterative method with exact solution per contact, a significant improvement over the popular PGS used in computer graphics, though still with no convergence nor accuracy guarantees.”
MethodDistributed Data-Driven Predictive Control for Multi-Agent Collaborative Legged Locomotion
Fawcett, R. T., Amanzadeh, L., Kim, J., Ames, A. D., & Hamed, K. A. (2023) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra48891.2023.10160914 · Cited by 27 · see citing works
“During the data collection procedure, the quadrupeds are commanded to walk around in RaiSim [44] using just the low-level controller (13), while random noise is injected into”
MethodHigh-speed control and navigation for quadrupedal robots on complex and discrete terrain
Kim, H., Oh, H., Park, J., KIM, Y. H., Youm, D., Jung, M., Lee, M., & Hwangbo, J. (2025) · Science Robotics · doi:10.1126/scirobotics.ads6192 · Cited by 27 · see citing works
“We generated 300 environments in parallel using the Raisim ( 49 ) physics simulation, with 4.2-second episodes, a 2 ms simulation time step, and a 0.01 s control time step.”
InfluentialSelf-citationDexPBT: Scaling up Dexterous Manipulation for Hand-Arm Systems with Population Based Training
Petrenko, A., Allshire, A., State, G., Handa, A., & Makoviychuk, V. (2023) · Robotics: Science and Systems · doi:10.15607/rss.2023.xix.037 · Cited by 25 · see citing works
Constrained stochastic optimal control with learned importance sampling: A path integral approach
Carius, J., Ranftl, R., Farshidian, F., & Hutter, M. (2021) · The International Journal of Robotics Research · doi:10.1177/02783649211047890 · Cited by 24 · see citing works
“For the purpose of forward-simulating stochastic roll-outs, we again use the RaiSim physics engine.”
MethodSelf-citationPower-Based Safety Layer for Aerial Vehicles in Physical Interaction Using Lyapunov Exponents
Cuniato, E., Lawrance, N., Tognon, M., & Siegwart, R. (2022) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2022.3176959 · Cited by 23 · see citing works
“V-B has been repeated multiple times with different gains in the Raisim simulation environment [23].”
MethodImitating and Finetuning Model Predictive Control for Robust and Symmetric Quadrupedal Locomotion
Youm, D., Jung, H., Kim, H., Hwangbo, J., Park, H.-W., & Ha, S. (2023) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2023.3320827 · Cited by 25 · see citing works
“2) Simulation: We use RaiSim [41] as our physics simulator.”
MethodSelf-citationAdaptiveON: Adaptive Outdoor Local Navigation Method for Stable and Reliable Actions
Liang, J., Weerakoon, K., Guan, T., Karapetyan, N., & Manocha, D. (2022) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2022.3229907 · Cited by 21 · see citing works
Probabilistic Contact State Estimation for Legged Robots using Inertial Information
Maravgakis, M., Argiropoulos, D.-E., Piperakis, S. M., & Trahanias, P. E. (2023) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra48891.2023.10161485 · Cited by 22 · see citing works
Fawcett, R. T., Pandala, A., Ames, A. D., & Hamed, K. A. (2021) · IEEE Control Systems Letters · doi:10.1109/lcsys.2021.3133198 · Cited by 22 · see citing works
Geometric Algebra for Optimal Control With Applications in Manipulation Tasks
Löw, T., & Calinon, S. (2023) · IEEE Transactions on Robotics · doi:10.1109/tro.2023.3277282 · Cited by 21 · see citing works
“These libraries include Raisim [57], Pinocchio [58], and KDL [59].”
BackgroundHierarchical Planning and Control for Box Loco-Manipulation
Xie, Z., Tseng, J., Starke, S., van de Panne, M., & Liu, C. K. (2023) · Proceedings of the ACM on Computer Graphics and Interactive Techniques · doi:10.1145/3606931 · Cited by 21 · see citing works
“RAISIM [Hwangbo et al. 2018] to simulate the rigid body dynamics and Proximal Policy Optimization [Schulman et al.”
MethodDeep reinforcement learning for real-world quadrupedal locomotion: a comprehensive review
Zhang, H., Li, H., & Wang, D. (2022) · Intelligence & Robotics · doi:10.20517/ir.2022.20 · Cited by 21 · see citing works
“Common robotic simulators, such as Pybullet and RaiSim, can only solve control-level simulations, but they are stretched for real-world simulations.”
BackgroundHybrid Bipedal Locomotion Based on Reinforcement Learning and Heuristics
Wang, Z., Wei, W., Xie, A., Zhang, Y., Wu, J., & Zhu, Q. (2022) · Micromachines · doi:10.3390/mi13101688 · Cited by 19 · see citing works
“The simulation environment is built with RaiSim [32].”
MethodCsomay-Shanklin, N., Cosner, R. K., Dai, M., Taylor, A. J., & Ames, A. D. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2105.01697 · Cited by 19 · see citing works
“The same controller (SS-QP) was deployed in the RaiSim (Hwangbo et al., 2018) simulation environment and on the AMBER-3M hardware platform, as seen in the supplementary video (vid).”
Design and Control of a Bio-Inspired Wheeled Bipedal Robot
Zhao, H., Yu, L., Qin, S., Jin, G., & Chen, Y. (2024) · IEEE/ASME Transactions on Mechatronics · doi:10.1109/tmech.2024.3449397 · Cited by 18 · see citing works
“Constraints (7) is the full dynamics, (37c) is the box constraints for the driven torques, (37d) is the linear approximation of the friction cone of each contact point defined in (13), while (37e) denotes the relaxed CLF condition defined in (30).”
Zuo, C., He, K., Shao, J., & Sui, Y. (2024) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra57147.2024.10610081 · Cited by 18 · see citing works
“Some recently developed physics engines [5], [20]–[22] are becoming popular for robotic simulation, and they are capable for efficient dynamic simulations that support contact interactions.”
BackgroundSemGrasp : Semantic Grasp Generation via Language Aligned Discretization
Li, K., Wang, J., Yang, L., Lu, C., & Dai, B. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-72627-9_7 · Cited by 21 · see citing works
Monte Carlo Tree Search Gait Planner for Non-Gaited Legged System Control
Amatucci, L., Kim, J.-H., Hwangbo, J., & Park, H.-W. (2022) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra46639.2022.9812421 · Cited by 17 · see citing works
Self-citationChen, J., Frey, J., Zhou, R., Miki, T., Martius, G., & Hutter, M. (2024) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2024.3455788 · Cited by 17 · see citing works
Self-citationWhole-Body Control of Series-Parallel Hybrid Robots
Mronga, D., Kumar, S., & Kirchner, F. (2022) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra46639.2022.9811616 · Cited by 16 · see citing works
Learning Forward Dynamics Model and Informed Trajectory Sampler for Safe Quadruped Navigation
Kim, Y., Kim, C., & Hwangbo, J. (2022) · Robotics: Science and Systems · doi:10.15607/rss.2022.xviii.069 · Cited by 16 · see citing works
Self-citationControl of Wheeled-Legged Quadrupeds Using Deep Reinforcement Learning
Lee, J., Bjelonic, M., & Hutter, M. (2022) · Lecture notes in networks and systems · doi:10.1007/978-3-031-15226-9_14 · Cited by 15 · see citing works
Self-citationPhysical factors that differentiate body kinematics between treadmill and overground walking
Jung, M., & Koo, S. (2022) · Frontiers in Bioengineering and Biotechnology · doi:10.3389/fbioe.2022.888691 · Cited by 14 · see citing works
“The dynamics simulation environment was created using RaiSim (Hwangbo et al., 2018), which is a multipurpose dynamics engine.”
MethodSafety-Critical Coordination for Cooperative Legged Locomotion via Control Barrier Functions
Kim, J., Lee, J., & Ames, A. D. (2023) · IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) · doi:10.1109/iros55552.2023.10341987 · Cited by 15 · see citing works
“We used RaiSim [27] for the simulation environment.”
MethodSelf-citationKoo, Y.-J., Hwangbo, J., & Koo, S. (2023) · Scientific Reports · doi:10.1038/s41598-023-49865-w · Cited by 17 · see citing works
“1.2 56 (Fig.”
MethodInfluentialSelf-citationMa, W.-L., Csomay-Shanklin, N., Kolathaya, S., Hamed, K. A., & Ames, A. D. (2021) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2021.3065174 · Cited by 15 · see citing works
“As a result, the local controller utilizing CCLFs renders stabilization of the given periodic gaits for walking, hopping and running on Vision 60 in RaiSim.”
MethodLearning Soccer Juggling Skills with Layer-wise Mixture-of-Experts
Xie, Z., Starke, S., Ling, H. Y., & van de Panne, M. (2022) · ACM SIGGRAPH Conference (SIGGRAPH) · doi:10.1145/3528233.3530735 · Cited by 15 · see citing works
“We use RAISIM [Hwangbo et al. 2018] as the physics simulator.”
MethodDistributed Quadratic Programming-Based Nonlinear Controllers for Periodic Gaits on Legged Robots
Kamidi, V. R., Kim, J., Fawcett, R. T., Ames, A. D., & Hamed, K. A. (2022) · IEEE Control Systems Letters · doi:10.1109/lcsys.2022.3167795 · Cited by 14 · see citing works
Learning Gait-conditioned Bipedal Locomotion with Motor Adaptation*
Wei, W., Wang, Z., Xie, A., Wu, J., Xiong, R., & Zhu, Q. (2023) · IEEE-RAS International Conference on Humanoid Robots (Humanoids) · doi:10.1109/humanoids57100.2023.10375167 · Cited by 15 · see citing works
“We build and evaluate our approach in Raisim [25], a cross-platform multi-body physics engine for robotics and AI.”
MethodCooperative Locomotion Via Supervisory Predictive Control and Distributed Nonlinear Controllers
Kim, J., & Hamed, K. A. (2021) · Journal of Dynamic Systems Measurement and Control · doi:10.1115/1.4052917 · Cited by 14 · see citing works
Learning Low-Frequency Motion Control for Robust and Dynamic Robot Locomotion
Gangapurwala, S., Campanaro, L., & Havoutis, I. (2023) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra48891.2023.10160357 · Cited by 13 · see citing works
“To support these points, we evaluate the performance of each of the individual blind π b :0 policies in RaiSim with unstructured rough terrain generated using Perlin noise [41] with maximum extrusion of 0 .”
InfluentialHigh-Fidelity Integrated Aerial Platform Simulation for Control, Perception, and Learning
Du, J., Wang, K., Fan, Y., Lai, G., & Yu, Y. (2025) · IEEE Transactions on Automation Science and Engineering · doi:10.1109/tase.2025.3555014 · Cited by 13 · see citing works
“A data interface has been designed for RaiSim [37], enhancing the simulator’s capacity for dynamic simulation of IAP platforms, thereby facilitating more precise dynamics and contact simulation.”
Legged Robot State Estimation with Invariant Extended Kalman Filter Using Neural Measurement Network
Youm, D., Oh, H., Choi, S., Kim, H., Jeon, S.-H., & Hwangbo, J. (2025) · Proceedings - IEEE International Conference on Robotics and Automation/Proceedings · doi:10.1109/icra55743.2025.11127971 · Cited by 18 · see citing works
“During the learning process, we sample trajectories through the locomotion policy in the simulation and train the NMN.”
Self-citationLearning Locomotion For Legged Robots Based on Reinforcement Learning: A Survey
Yue, J. (2020) · International Conference on Electrical Engineering and Control Technologies (CEECT) · doi:10.1109/ceect50755.2020.9298680 · Cited by 13 · see citing works
“Vassilios Tsounis et al.[30] developed Raisim[45] based on mujoco.”
BackgroundSim-to-Real Transfer of Bolting Tasks with Tight Tolerance
Son, D., Yang, H., & Lee, D. (2020) · IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) · doi:10.1109/iros45743.2020.9341644 · Cited by 13 · see citing works
“We then use these W-space contact points into the contact solver (e.g. [10], [15], [16]).”
MethodInfluentialLearning to adapt through bio-inspired gait strategies for versatile quadruped locomotion
Humphreys, J., & Zhou, C. (2025) · Nature Machine Intelligence · doi:10.1038/s42256-025-01065-z · Cited by 14 · see citing works
“Overall, on smooth low-friction floor π uni G is able to achieve a 15% and 24% reduction in CoT compared to trotting and running gaits and a reduction of 18% and 30% on uneven grass.”
Learning Perceptual Locomotion on Uneven Terrains Using Sparse Visual Observations
Acero, F., Yuan, K., & Li, Z. (2022) · IEEE Robotics and Automation Letters · doi:10.1109/lra.2022.3188108 · Cited by 12 · see citing works
Sathya, A. S., Bruyninckx, H., Decré, W., & Pipeleers, G. (2023) · IEEE Transactions on Robotics · doi:10.1109/tro.2023.3335652 · Cited by 12 · see citing works
“OSIM is particularly useful for resolving inequality constraints (also called unilateral constraints), because an inequality constraint becoming inactive can be easily handled by removing the corresponding row and column of the inverse OSIM and efficiently updating the factorization [55].”
BackgroundMAGIC VFM -Meta-Learning Adaptation for Ground Interaction Control With Visual Foundation Models
Lupu, E. S., Xie, F., Preiss, J. A., Alindogan, J., Anderson, M., & Chung, S. (2024) · IEEE Transactions on Robotics · doi:10.1109/tro.2024.3475212 · Cited by 12 · see citing works
“This capability is achieved by leveraging high-fidelity models in robotics simulators [51], [52], imitation learning from human demonstrations or techniques to bridge the sim-to-real gap of the learned policy [14].”
BackgroundReinforcement learning in robotic systems : A review on sim-to-real transfer
Tiwari, R., Khapre, S. P., & Singh, A. (2026) · Robotics and Autonomous Systems · doi:10.1016/j.robot.2025.105327 · Cited by 11 · see citing works
- Authors
- Hwangbo, J., Lee, J., & Hutter, M.
- Affiliations
- ETH Zurich (CH)
- Venue
- IEEE Robotics and Automation Letters
- Published
- 2018-04
- Volume
- 3
- Issue
- 2
- Pages
- 895-902
- ISSN
- 2377-3766 (ISSN-L)
- 2377-3774
- OpenAlex
- W2779458783
- Cited by
- 243
- References
- 19
- Citations by year
- 2026: 20 · 2025: 43 · 2024: 41 · 2023: 42 · 2022: 49 · 2021: 33 · 2020: 12 · 2019: 2 · 2018: 1
- FWCI
- 7.43
- Type
- journal article
- Language
- en
- Open access
- green
- hdl.handle.net
- Locations
- IEEE Robotics and Automation Letters (publishedVersion)
- Repository for Publications and Research Data (ETH Zurich) (submittedVersion, other-oa)
- ETH Zürich Research Collection
- Funders
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
- National Centre of Competence in Research Robotics and Project (200021-166232)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Dynamics and Control of Mechanical Systems
- Physical Sciences › Engineering › Biomedical Engineering › Robotic Locomotion and Control
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Contact Mechanics and Variational Inequalities
- Keywords
- contact dynamics, bisection method, computational efficiency
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