That question has fascinated me since childhood. At first it felt like a dream, but over time I realized it was more than a curiosity. It became a framework for understanding the world: not just anticipating what will happen, but asking why it happens and learning how to shape outcomes through knowledge, science, and design.

This passion drew me to dynamics, the science of motion, and optimization, the art and science of finding the most effective path forward in a world full of trade-offs. Together, they became the foundation of my work in robotics, where every challenge is a balance between theory and reality.

Lessons in Focus and Resilience

Before robotics, my early experiences in competitive sports taught me the importance of precision, discipline, and resilience. Those lessons shaped the way I approached research. Success does not come from avoiding obstacles but from moving through them, learning at each step. This mindset became vital as I pursued increasingly ambitious projects in robotics.

Walking Robots and Early Discoveries

At Vanderbilt University, I was challenged to build a walking robot independently. What could have been an impossible task became a defining experience. The project was filled with setbacks, redesigns, and breakthroughs, but eventually the robot took its first steps. That moment was unforgettable, but even more important were the insights I gained about how to connect abstract theory with physical systems.

Crossing Disciplines for Innovation

My journey continued at the University of Edinburgh and the German Aerospace Center, where I worked across mechanical engineering and informatics. Our team developed control methods for the complex DLR Hand-Arm System, creating robotic motion that was remarkably lifelike. This work was later recognized with the IEEE Transactions on Robotics Best Paper Award, a milestone that reinforced my belief that innovation thrives at the intersection of disciplines.

Building a Foundation for the Next Generation

In Singapore, I had the opportunity to help establish a new university. It was a chance to think not only about research but also about how to shape an institution from the ground up. Alongside collaborators, I developed compliant actuators, novel robotic hardware, and real-time optimal control methods. These projects demonstrated how adaptive machines could better navigate the unpredictability of the real world. Building something that would outlast me, a place where innovation could grow, was as meaningful as the research itself.

The Boundaries of Robotics

Returning to Vanderbilt, my work turned toward human augmentation. I developed a theoretical framework for wearable devices that can enhance human performance without external power. The bicycle was my inspiration, a simple yet powerful example of how mechanics can amplify human capability. This research was published in Science Advances and featured in The Guardian and The Conversation US, reaching audiences beyond academia.

As a culmination of my academic work in the foundations of robotics, I received the National Science Foundation CAREER Award. For me, it was not a recognition of past achievements but a validation of the belief that rigorous, interdisciplinary science can expand the possibilities of robotics.

Looking Ahead: Beyond the Lab

Today, my focus is on advancing robotics beyond papers and prototypes to systems that perform reliably in real-world conditions. The future of robotics will not be defined solely by faster processors or larger datasets. It will be defined by how we design machines that adapt, learn, and operate under the constraints of unstructured environments.

My current work centers on creating adaptive hardware that improves performance, developing intelligent control methods that close the gap between simulation and deployment, and designing next-generation robotics platforms that translate theoretical advances into capability at scale.

What began as a childhood fascination with predicting the future has become a mission: to create it. That mission continues today, driven by the same curiosity and determination that set me on this path, and by a commitment to advance robotics toward systems that are adaptive, reliable, and transformative.

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Dynamics is a branch of Mechanics that studies the motion of objects under the influence of forces. It is fundamental to engineering, bridging theoretical principles from physics with practical applications to predict the motion of systems such as rockets, cars, and robots. Introduction to Dynamics is the first course on this topic, studied by engineering, math, and science students.

My Expertise

I specialize in teaching Mechanics, focusing on both theoretical and computational aspects of:

• Newtonian Mechanics: Principles governing motion and forces.
• Lagrangian Mechanics: Energy-based methods for dynamic systems.
• Hamiltonian Mechanics: Advanced formulations that extend dynamics to optimal control.

My goal is to ensure students develop both a solid theoretical foundation and effective problem-solving skills.

Teaching

My teaching focuses on clarity and engagement, helping students navigate complex topics with confidence. Key features include:

• Bite-Sized Video Lectures: Pre-class videos introducing core concepts and examples.
• Interactive Lectures: Expanding on pre-class material with deeper theoretical insights.
• Emphasizing Mathematical Formalism: Leveraging formal methods to overcome the limitations of intuition-based problem-solving.

Why Study Dynamics?

Consider a simple question: If you drop a pen from your desk, what truly determines its trajectory? The intuitive answer might involve gravity, but a rigorous approach demands an understanding of differential equations, constraints, and possibly even perturbation theory. What if the pen is in freefall on the Moon? Or inside a rotating space station? The same fundamental principles govern everything from planetary orbits to the stabilization of legged robots.

This course will challenge you to refine your thinking, stripping away assumptions and replacing them with precise models that describe reality with mathematical accuracy. The tools you develop here will extend far beyond the classroom—whether you are designing autonomous drones, optimizing mechanical systems, or advancing theoretical physics.

Dynamics is not about memorization or following recipes; it is about learning to see structure in motion and gaining the ability to predict and control it. If that is the kind of challenge you are looking for, you are in the right place.

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Optimization and Optimal Control are essential tools for designing systems that operate efficiently while minimizing cost, time, or resources. These methods provide a systematic approach to decision-making in engineering and science, enabling solutions for problems ranging from fuel-efficient spacecraft trajectories to real-time energy optimization in robotics and improving machine learning models. By translating abstract mathematical principles into real-world applications, this course equips you with the ability to bridge theory and practice in optimization.

My Expertise

I specialize in Optimization and Optimal Control, focusing on both theoretical foundations and computational methods related to:

• Non-linear Programming: Techniques for solving static optimization problems—minimizing functions subject to constraints.
• Calculus of Variations: Methods for solving dynamic optimization problems—minimizing functionals subject to constraints.
• Optimal Control:
  • Maximum Principle: A powerful framework to solve optimal control problems and determine feedforward control inputs.
  • Dynamic Programming: A general framework to solve optimal control problems and derive feedback control laws or control policies.

Teaching

I use a teaching approach that makes complex concepts in optimization and control accessible and engaging. My method emphasizes:

• Step-by-Step Analogies: Building from simple, familiar problems like minimizing a quadratic function to more complex problems.
• Interactive Learning: Encouraging active participation, from hands-on MATLAB coding to solving real-world engineering problems.
• Bridging Disciplines: Linking optimization to applications in robotics, aerospace, and mechanics for a holistic understanding.

Why Study Optimization and Optimal Control?

Every engineering challenge ultimately involves making decisions—how to allocate resources, how to design efficient systems, and how to control them in real time. Optimization is the language that enables us to make these decisions systematically.

Consider a simple problem: if you are launching a spacecraft, should you aim for the shortest path or the most energy-efficient trajectory? The naive answer might be to take a direct route, but an optimal control perspective would reveal trade-offs between fuel, time, and orbital mechanics that must be mathematically optimized. The same principles apply whether you are designing an autonomous drone, optimizing neural network hyperparameters, or developing an exoskeleton to enhance human mobility.

Optimization is not about guessing or intuition—it is about constructing solutions with mathematical precision and algorithmic efficiency. If you are interested in developing the skills to systematically tackle the most complex problems in engineering and science, this course will give you the tools to do exactly that.

The post Optimization and Optimal Control appeared first on David Braun.

Unpowered mechanical devices are already used for human performance enhancement; a person can travel much farther and faster on a bicycle than they can on foot. The exoskeletons we work on go beyond such familiar energetically passive devices by integrating actively controlled energy storage, in the form of springs of controllable stiffness. In our robots, the amount of energy stored in a spring of controllable stiffness can be decoupled from the deformation of the spring. This new capability could advance human and robot mobility.

Running with a Variable Stiffness Spring Leg

Explore the thrilling potential of human speed with our analysis of top speeds in running, ice skating, cycling, and running enhanced by variable stiffness spring legs. The figure below plots these top speeds against the time available for the legs to supply energy in each motion. Naturally, in running, legs supply energy only when in contact with the ground—about 20% of the time and not during flight. However, with the aid of a variable stiffness spring, a human could theoretically supply energy for up to 96% of the leg swing. Even at a lower efficiency of 60%, this innovative exoskeleton could enable speeds of 18 m/s, a 50% increase over the natural running top speed of 12 m/s, as posited by our theoretical model.

• A. Sutrisno and D. Braun, How to run 50% faster without external energy, Science Advances, 6(13), eaay1950, 2020.

The figure below shows the method of augmented running using a variable stiffness spring where the human compresses the spring when the leg is in the air while the spring releases the energy stored when the leg touches the ground.

The video below shows a thought experiment of a 100-meter race, where a world-class athlete augmented with the variable stiffness spring leg exoskeleton races against the 9.58 seconds world record set by the Jamaican sprinter Usain Bolt in 2009.

Rethinking Assistive Robotics with Floating Spring Legs

Imagine a spring that adapts to the task at hand, delivering more force without consuming additional energy. The floating spring leg is an innovative variable stiffness mechanism that achieves this by changing stiffness while conserving energy. Demonstrated through a simulated sit-to-stand task, the leg transitions from a low stiffness configuration for energy storage to a high stiffness configuration for force amplification, enabling movements that are otherwise impossible with conventional springs. With applications ranging from assistive devices for the elderly to industrial robots, this technology paves the way for energetically efficient, mechanically adaptive systems.

Customizing Assistance with Variable Stiffness Springs

A novel robot joint design enables users to adjust joint stiffness with ease, much like shifting gears on a bicycle. The design allows for adjusting joint stiffness to suit different tasks or user needs, such as walking, running, or other movements. The concept is demonstrated through a leg swing experiment, where hip stiffness was adjusted seamlessly during continuous leg motion. This innovation has the potential to optimize performance in applications such as prosthetics and robotic exoskeletons, offering users greater adaptability and efficiency across activities.

• C. Mathews and D.J. Braun, Design of a Variable Stiffness Spring with Human-Selectable Stiffness, IEEE International Conference on Robotics and Automation, London, United Kingdom, pp. 7385-7390, 2023.

Energy Accumulation via Repeated Squatting

A novel variable stiffness spring design makes it possible to store significant amounts of energy by applying small, repeated forces. Unlike traditional springs, which demand large forces for substantial energy storage, this mechanism decouples the maximum force applied from the energy stored. The concept is demonstrated using an assistive robotic leg that accumulates energy as the user performs repeated squats. This innovation has the potential to unlock new energy storage capabilities for applications such as assistive devices and robotic exoskeletons, paving the way for more efficient and versatile robotic systems.

Walking with a Variable Stiffness Spring Leg

This work establishes the theoretical limits of a human-driven robot exoskeleton to reach fast walking speeds with a heavy load. We predict that a variable stiffness spring exoskeleton could enable the user to reach race-walking speeds (4m/s) while carrying one extra body weight. 

• T. Zhang and D. Braun, Theory of Fast Walking with Human-Driven Load-Carrying Robot Exoskeletons, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 30, 1971-1981, 2022.

Human Driven Compliant Transmission Mechanism

Energetically-passive robot exoskeletons, mimicking the function of the bicycle, could enable humans to reach previously unprecedented mobility. However, energetically passive robot exoskeletons require a sophisticated mechanism to enable humans to supply energy. In this research, we develop a new type of human-driven compliant transmission mechanism that could enable humans to supply energy when the leg is in the air, store the supplied energy, and release the stored energy when the leg is on the ground.

• T. Zhang and D. Braun , Human Driven Compliant Transmission Mechanism, IEEE International Conference on Robotics and Automation, pp. 7094-7099, Xi’an, China, 2021.

Variable Stiffness Springs for Human Augmentation

Variable stiffness springs are devices that promote a novel means of actuation; they provide spring stiffness modulation at a low energy cost, without deliberately doing net mechanical work. This type of compliant actuator may be used for human augmentation to complement co-contracted antagonistic muscles, and as such, reduce muscle activity and metabolic energy cost. Despite the theoretical appeal of this concept, the practical use of variable stiffness springs remains challenging. This is particularly true for human augmentation which requires a compact, lightweight, and portable variable stiffness spring design. The figure below shows one of our compact, lightweight, and portable variable stiffness spring designs.

• H.-F. Lau, A. Sutrisno, T.H. Chong, and D. Braun, Stiffness Modulator: A Novel Actuator for Human Augmentation, IEEE International Conference on Robotics and Automation, pp. 7742-7748, May, Brisbane, Australia, 2018.
• D. Braun, S. Apte, O. Adiyatov, A. Dahiya, and N. Hogan, Compliant Actuation for Energy Efficient Impedance Modulation, IEEE International Conference on Robotics and Automation, pp. 636-641, May, Stockholm, Sweden, 2016.

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Custom-designed Magnetic Spring Actuators

Robots face a persistent challenge: motors consume energy not only when performing useful work but also when holding a position or braking, where no meaningful work is done. This inefficiency becomes especially critical in repetitive tasks like pick-and-place operations or oscillatory motions, where a significant amount of energy is wasted for only a small amount of useful work. Our research introduces a novel solution—combining motors with custom-designed magnetic springs that leverage non-uniform magnetic fields optimized for specific tasks. These actuators are precisely tailored to match the dynamics of individual robotic operations, significantly reducing the motor’s torque and energy demands. Experiments show up to 50% energy savings, marking a significant step toward more energy-efficient robotics. By addressing this critical limitation, these actuators open new possibilities for deploying sustainable, high-performance robots in a broad range of industrial settings while reducing the energy consumption of robots across various industries.

• Y. Y. Fu, A. U. Kilic, and D. J. Braun, Energy Minimization using Custom-Designed Magnetic-Spring Actuators, IEEE International Conference on Intelligent Robots and Systems, 2024.

Parallel Variable Stiffness Actuators (PVSA)

In this research, we explore the design of a new type of compliant actuator named the Parallel Variable Stiffness Actuator (PVSA) which consists of a variable stiffness spring placed in parallel with a direct-drive motor. Parallel variable stiffness actuators provide (i) high-fidelity force control and (ii) controllable energy storage, as they inherit the benefits of direct-drive motors and variable stiffness springs. We present a compact design of the PVSA using a flat motor connected to an adjustable mechanical advantage torsional spring. We show that PVSAs are (1) not subject to the fundamental force control bandwidth limitation of series elastic and variable stiffness actuators, and (2) enable resonant energy accumulation despite the limited deformation of the spring and the constrained motion of the load attached to the actuator. The latter differentiates parallel variable stiffness actuators from fixed-stiffness parallel elastic actuators. PVSAs may be used with smaller direct-drive motors to match the peak power of larger motors without compromising force control fidelity. PVSAs may be used in industrial robots to provide both precise manipulation and zero-energy cost weight-bearing. PVSAs may be also used to implement resonant forcing under joint angle limitations in walking, jumping, running, and swimming robots, or robotic exoskeletons used to augment human motion.

• M. W. Chase and D. Braun, Design of Parallel Variable Stiffness Actuators, IEEE Transactions on Robotics, 2022.

Variable Stiffness Springs (VSS)

Theoretical studies suggest and experimental evidence confirms that maintaining and changing human joint stiffness by coactivated antagonistic muscles are metabolically expensive, even if muscles do not perform net mechanical work. Based on this observation, we conjecture that effective human augmentation can be achieved by actuators operated in parallel to human joints, even if these actuators only supplement joint stiffness without doing net mechanical work. In this research, we developed a variable-length leaf-spring actuator capable of large-range stiffness modulation. The key feature of the actuator is that it provides intrinsically low-energy-cost stiffness modulation even for large output deflection, by keeping the force on the driving motor low. Variable stiffness actuators use two motors to provide both stiffness and equilibrium position modulation as they are designed to do net mechanical work. The proposed actuator conceptually differs from variable stiffness actuators because first, it uses a single motor to only provide stiffness modulation, second, it does not provide equilibrium position modulation, and third, unless externally loaded, it cannot do net mechanical work. Using this actuator, we demonstrate stiffness augmentation during human–machine collaboration in challenging postural stabilization and weight-bearing tasks. Our results indicate that the proposed actuator can be used to complement a biological system by restoring or extending its functionality with low energy cost, and that variable stiffness spring actuators could effectively augment humans by doing no or a limited amount of mechanical work.

• D. Braun, V. Chalvet, T.H. Chong, S.S. Apte, and N. Hogan, Variable Stiffness Spring Actuators for Low Energy Cost Human Augmentation, IEEE Transactions on Robotics, vol. 35, no. 6, pp. 1435-1449, 2019.

Positive Negative Stiffness Actuators (PNSA)

Compliant actuators typically possess a tunable positive stiffness characteristic to generate restoring force upon displacement. These actuators are similar to adaptive springs that use two independent motor units or closed-loop control to change both their equilibrium position and stiffness. The introduction of negative stiffness, in combination with tunable positive stiffness, may reduce the complexity and extend the capability of these actuators in unexpected ways. In this research, we explore novel designs of compliant actuators that employ a passive negative stiffness mechanism in conjunction with an effectively tunable positive stiffness mechanism. We show that such an actuator can enable open-loop stiffness modulation and equilibrium position control using a single motor unit, as opposed to more conventional variable stiffness and series elastic actuators.

• D. Braun, V. Chalvet, and A. Dahiya Positive-Negative Stiffness Actuators, IEEE Transactions on Robotics, vol. 35, no. 1, pp. 162-173, 2019.

Computational Design of Variable Stiffness Actuators

Prior research on robot actuator design identified significant limitations of experience-based design methods used to develop variable stiffness mechanisms, actuators, and robots. In this research, we develop an optimization-based computational framework for the design of variable stiffness mechanisms, actuators, and robots. The core ingredient of this framework is the mathematical formulation of the design problem which is computationally solved to identify intrinsically low-power variable stiffness mechanism designs.

• V. Chalvet and D.J. Braun, Algorithmic Design of Low Power Variable Stiffness Mechanisms, IEEE Transactions on Robotics, vol. 33, no. 6, pp. 1508-1515, 2017.
• V. Chalvet and D.J. Braun, Criterion for the Design of Low Power Variable Stiffness Mechanisms, IEEE Transactions on Robotics, vol. 33, no. 4, pp. 1002-1010, 2017.

The post Mechanically Adaptive Actuators to Advance Robot Autonomy appeared first on David Braun.

We promote the optimization of robot performance via real-time experiments guided by dedicated teacher robots, instead of optimizing system performance guided by uncertain model-based predictions and measured data. The method we develop is where robots are used to develop controllers for robots in a somewhat analogous way as parents teach their children. The technique may deliver a transformative approach to control engineering systems, for which obtaining high-fidelity models is challenging while gathering training data is time-consuming.

”Robots Teaching Robots” as a New Control Paradigm

Optimal control is a branch of control theory that has the potential to revolutionize the creation of intelligent engineering systems – assistive robots, industrial robots, medical robots – that can improve by repeated experience, somewhat similar to humans. There are many optimal control techniques to control engineering systems. However, most currently available techniques require high-fidelity models or a large amount of measured data to mitigate the so-called simulation-reality gap; the gap between the optimal robot performance predicted by computer simulations and the non-optimal robot performance observed in real engineering applications.

We conduct fundamental research to close the simulation-reality gap when optimal control is applied to engineering systems. Model-based optimal control techniques enable efficient computation but they are subject to conservative control performance. Data-driven optimal control techniques mitigate the detrimental effect of uncertain models, but to do so, they require a large amount of training data. Therefore, scientific barriers must be overcome to realize the full application potential of optimal control techniques.

We promote the optimization of system performance via real-time experiments guided by dedicated teacher robots, instead of optimizing system performance guided by uncertain model-based predictions and measured data. Our research aims to address the knowledge gap that limits the potential and theoretical promise of optimal control theory when applied to engineering systems.

Real-time Optimal Control of Robots

Optimal controllers computed using inexact model information lead to significant performance degradation. One way to avoid this limitation is to use model-free optimal control. However, model-free optimal control approaches require a large amount of training data. The goal of this research is to introduce an optimal control formulation that uses inexact models to speed up the computation and measured data to avoid the simulation reality gap introduced by inexact model-based future prediction. The result is a real-time – half model-based and half data-driven – optimal control formulation which is suitable for real-time hardware-in-the-loop control. Experiments on a three-link direct drive torque-controlled robot demonstrate the benefits of the proposed hardware-in-the-loop optimal control method.

• Y. Chen, Y. Li and D. J. Braun, Data-Driven Iterative Optimal Control for Switched Dynamical Systems, IEEE Robotics and Automation Letters, vol. 8, no. 1, pp. 296-303, 2023.
• Y. Chen and D.J. Braun, Iterative Online Optimal Feedback Control, IEEE Transactions on Automatic Control, vol. 66, no. 2, pp. 566-580, 2021.
• Y. Chen and D.J. Braun, Hardware-in-the-Loop Iterative Optimal Feedback Control without Model-based Future Prediction, IEEE Transactions on Robotics, vol. 35, no. 6, pp. 1419-1434, 2019.

Optimal Control of Compliantly Actuated Robots

Compliant actuation is a biologically inspired actuation concept which promises mechanical adaptability and unseen agility for new-generation robots. However, compliant actuation introduces non-linearity, actuation constraints, and control redundancy which make robots challenging to control. In this research, we investigate how impedance control strategies emerge from the first principles of optimality, and how these strategies can be used to actuate compliant variable impedance robots. We used computational optimal control to make the ”DLR David” robot perform a highly dynamic ball-throwing task – similar to javelin throwing at the Olympics –which requires precise timing and perfect execution. The video below shows the result. The behavior of the robot is surprising because it is predicted by numerical optimization without the use of human data or expert demonstration.

• D.J. Braun, F. Petit, F. Huber, S. Haddadin, P. van der Smagt, A. Albu-Schaffer, and S. Vijayakumar, Robots Driven by Compliant Actuators: Optimal Control under Actuation Constraints, IEEE Transactions on Robotics, vol. 29, no. 5, pp. 1085-1101, 2013.

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Reverse Engineering Impedance Controllers from Human Locomotion Data

The objective of this research is to investigate whether a small number of sequentially composed multivariable linear controllers can be used to recover a defining relation between the joint torques, angles, and velocities hidden in the walking data of multiple human subjects. Using the data of seven healthy subjects, we show that the aforementioned relation can be well approximated by four sequentially composed and independently activated multivariable linear controllers. We further show that each controller is associated with one of the four phases of the gait cycle, separated by toe-off and heel-strike. The proposed controller generalizes previously developed multiphase single variable, and single phase multivariable controllers, to a multiphase multivariable controller that better explains the walking data of multiple subjects, and better generalizes to new subjects. Our result provides strong support to extend previously developed decoupled single-joint single-variable controllers to coupled multi-joint multivariable controllers for the control of walking robots as well as human assistive and augmentation devices.

• E.S. Altinkaynak and D.J. Braun, Multiphase and Multivariable Linear Controllers that Account for the Joint Torques in Normal Human Walking, IEEE Transactions on Biomedical Engineering, vol. 67, no. 6, pp. 1573-1584, 2020.

Switching Impedance Control for Legged Locomotion

We hypothesized that an optimization-based generalization of single-joint switching impedance controllers used for limb coordination in prosthetic devices is capable of resolving the complex coordination task underlying bipedal locomotion including not only limb coordination in prosthetic devices but also upper-body stabilization in walking robots. Numerical simulations and experiments conducted on an anthropomorphic biped robot provide strong support for this hypothesis.

• D.J. Braun, J.E. Mitchell, and M. Goldfarb, Actuated Dynamic Walking in a Seven-Link Biped Robot, IEEE/ASME Transactions on Mechatronics, vol. 17, no. 1, pp. 147-156, 2012.
• D.J. Braun and M. Goldfarb, A Control Approach for Actuated Dynamics Walking in Biped Robots, IEEE Transactions on Robotics, vol. 25, no. 6, pp. 1292-1303, 2009.

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Modeling and Simulating Robot Motion using switching DAEs

We present a differential-algebraic formulation with switching constraints to model the non-smooth dynamics of robotic systems subject to changing constraints and multiple impacts. The formulation combines a single structurally simple governing equation, a set of switching kinematic constraints, and the plastic impact law, to represent the dynamics of robots that interact with their environment.

• Y. Li, H. Yu, and D.J. Braun, Algorithmic Resolution of Multiple Impacts in Non-smooth Mechanical Systems with Switching Constraints, IEEE International Conference on Robotics and Automation, pp. 7639-7645, Montreal, QC, Canada, 2019.

Modeling and Simulating Robot Motion using DAEs

A large class of constrained dynamical systems can be modeled using differential algebraic equations (DAEs). Unlike ordinary differential equations (ODEs), which can be accurately integrated with explicit numerical methods (for example the Runge-Kutta method), accurate numerical integration of differential-algebraic equations requires sophisticated implicit integration methods. In this research, we develop explicit numerical integration algorithms which are easy to implement and can be used to accurately integrate differential algebraic equations. We assumed that the numerical solution is error contaminated, such that, neither the kinematic constraints nor any type of motion invariant, for example, an energy conservation law, can be exactly satisfied by a numerical solution. Using this assumption, we perform systematic mathematical derivations to find novel numerical error correction terms for accurate numerical integration of the differential-algebraic equations of motion.

• D.J. Braun and M. Goldfarb, Simulation of Constrained Mechanical Systems – Part I: An Equation of Motion, ASME Journal of Applied Mechanics, vol. 79, issue 4, 041017, 2012.
• D.J. Braun and M. Goldfarb, Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration, ASME Journal of Applied Mechanics, vol. 79, issue 4, 041018, 2012.
• D.J. Braun, M. Goldfarb, Elimination of Constrained Drift in the Numerical Simulation of Constrained Dynamical Systems, Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 37-40, pp. 3151-3160, 2009.

Our method is easy to implement and provides significant improvement in numerical accuracy when tested against the analytical solution of simple case-study problems or the numerical solution of complex problems modeled using ordinary differential equations. The test example below demonstrates the relation between the integrator we developed to precisely solve differential algebraic equations and the Runge-Kutta method which can only be used to precisely solve differential equations.

The post Physics-based Numerical Simulation Methods for Predicting Robot Behavior appeared first on David Braun.

Yixi Chen, Evangelos Chatziandreou, Chase Mathews, Beau Johnson, and David J. Braun, Design of a Parallel Elastic Actuator with Continuously Adjustable Equilibrium Position, IEEE Robotics and Automation Letters, vol. 11, no. 6, pp. 6512-6519, 2026.

We present an adjustable-equilibrium parallel elastic actuator (AE-PEA) that combines a large direct-drive motor (DDM) and a 3D printed torsional spring with a small motor that can continuously adjust the equilibrium position of the actuator. We demonstrate that the AE-PEA achieves high torque control bandwidth like direct-drive actuators (DDAs), energy storage and shock mitigation like parallel elastic actuators (PEAs), and equilibrium position adjustment similar to series elastic actuators (SEAs). Owing to these features, we foresee the benefit of AE-PEAs in robot joints performing a variety of static load bearing and dynamic oscillatory motion.

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Y. Chen, Y. Li and D.J. Braun, Learning Finite-Horizon Optimal Control with Unknown Control-Affine Dynamics, Systems & Control Letters, vol. 203, no. 106161, 2025.

This paper introduces a model-free method for designing optimal controllers over a fixed time horizon. Instead of relying on a detailed system model, the method adapts simple linear rules that learn to approximate the optimal control policy. The authors prove that the process converges to the true optimal solution and demonstrate it with a numerical example.

Why it matters: Many real-world systems are too complex to model accurately. This method learns high-performance control strategies directly from interaction, reducing dependence on precise models and advancing applications in robotics, automation, and beyond.

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