Ismail Uyanik

Experimental Validation of a Feed-Forward Predictor for the Spring-Loaded Inverted Pendulum Template

Widely accepted utility of simple spring-mass models for running behaviors as descriptive tools, as well as literal control targets, motivates accurate analytical approximations to their dynamics. Despite the availability of a number of such analytical predictors in the literature, their validation has mostly been done in simulation, and it is yet unclear how well they perform when applied to physical platforms. In this paper, we extend on one of the most recent approximations in the literature to ensure its accuracy and applicability to a physical monopedal platform. To this end, we present systematic experiments on a well-instrumented planar monopod robot, first to perform careful identification of system parameters and subsequently to assess predictor performance. Our results show that the approximate solutions to the spring-loaded inverted pendulum dynamics are capable of predicting physical robot position and velocity trajectories with average prediction errors of 2% and 7%, respectively. This predictive performance together with the simple analytic nature of the approximations shows their suitability as a basis for both state estimators and locomotion controllers.

Adaptive control of a spring-mass hopper

Practical realization of model-based dynamic legged behaviors is substantially more challenging than statically stable behaviors due to their heavy dependence on second-order system dynamics. This problem is further aggravated by the difficulty of accurately measuring or estimating dynamic parameters such as spring and damping constants for associated models and the fact that such parameters are prone to change in time due to heavy use and associated material fatigue. In this paper, we present an on-line, model-based adaptive control method for running with a planar spring-mass hopper based on a once-per-step parameter correction scheme. Our method can be used both as a system identification tool to determine possibly time-varying spring and damping constants of a miscalibrated system, or as an adaptive controller that can eliminate steady-state tracking errors through appropriate adjustments on dynamic system parameters. We present systematic simulation studies to show that our method can successfully accomplish both of these tasks.

Identification of a vertical hopping robot model via harmonic transfer functions

A common approach to understanding and controlling robotic legged locomotion is the construction and analysis of simplified mathematical models that capture essential features of locomotor behaviours. However, the representational power of such simple mathematical models is inevitably limited due to the non-linear and complex nature of biological locomotor systems. Attempting to identify and explicitly incorporate key non-linearities into the model is challenging, increases complexity, and decreases the analytic utility of the resulting models. In this paper, we adopt a data-driven approach, with the goal of furnishing an input–output representation of a locomotor system. Our method is based on approximating the hybrid dynamics of a legged locomotion model around its limit cycle as a Linear Time Periodic (LTP) system. Perturbing inputs to the locomotor system with small chirp signals yield the input–output data necessary for the application of LTP system identification techniques, allowing us to estimate harmonic transfer functions (HTFs) associated with the local LTP approximation to the system dynamics around the limit cycle. We compare actual system responses with responses predicted by the HTF, providing evidence that data-driven system identification methods can be used to construct models for locomotor behaviours.

Toward data-driven models of legged locomotion using harmonic transfer functions

There are limitations on the extent to which manually constructed mathematical models can capture relevant aspects of legged locomotion. Even simple models for basic behaviours such as running involve non-integrable dynamics, requiring the use of possibly inaccurate approximations in the design of model-based controllers. In this study, we show how data-driven frequency domain system identification methods can be used to obtain input-output characteristics for a class of dynamical systems around their limit cycles, with hybrid structural properties similar to those observed in legged locomotion systems. Under certain assumptions, we can approximate hybrid dynamics of such systems around their limit cycle as a piecewise smooth linear time periodic system (LTP), further approximated as a time-periodic, piecewise LTI system to reduce parametric degrees of freedom in the identification process. In this paper, we use a simple one-dimensional hybrid model in which a limit-cycle is induced through the actions of a linear actuator to illustrate the details of our method. We first derive theoretical harmonic transfer functions (HTFs) of our example model. We then excite the model with small chirp signals to introduce perturbations around its limit-cycle and present systematic identification results to estimate the HTFs for this model. Comparison between the data-driven HTFs model and its theoretical prediction illustrates the potential effectiveness of such empirical identification methods in legged locomotion.

Extending the lossy Spring-Loaded Inverted Pendulum model with a slider-crank mechanism

Spring Loaded Inverted Pendulum (SLIP) model has a long history in describing running behavior in animals and humans as well as has been used as a design basis for robots capable of dynamic locomotion. Anchoring the SLIP for lossy physical systems resulted in newer models which are extended versions of original SLIP with viscous damping in the leg. However, such lossy models require an additional mechanism for pumping energy to the system to control the locomotion and to reach a limit-cycle. Some studies solved this problem by adding an actively controllable torque actuation at the hip joint and this actuation has been successively used in many robotic platforms, such as the popular RHex robot. However, hip torque actuation produces forces on the COM dominantly at forward direction with respect to ground, making height control challenging especially at slow speeds. The situation becomes more severe when the horizontal speed of the robot reaches zero, i.e. steady hoping without moving in horizontal direction, and the system reaches to singularity in which vertical degrees of freedom is completely lost. To this end, we propose an extension of the lossy SLIP model with a slider-crank mechanism, SLIP-SCM, that can generate a stable limit-cycle when the body is constrained to vertical direction. We propose an approximate analytical solution to the nonlinear system dynamics of SLIP-SCM model to characterize its behavior during the locomotion. Finally, we perform a fixed-point stability analysis on SLIP-SCM model using our approximate analytical solution and show that proposed model exhibits stable behavior in our range of interest.

A high frequency 3D LiDAR with enhanced measurement density via Papoulis-Gerchberg

Light Detection and Ranging (LiDAR) devices are gaining more importance for obtaining sensory information in mobile robot applications. However, existing solutions in literature yield low frequency outputs with huge measurement delay to obtain 3D range image of the environment. This paper introduces the design and construction of a 3D range sensor based on rotating a 2D LiDAR around its pitch axis. Different than previous approaches, we adjust our scan frequency to 5 Hz to support its application on mobile robot platforms. However, increasing scan frequency drastically reduces the measurement density in 3D range images. Therefore, we propose two post-processing algorithms to increase measurement density while keeping the 3D scan frequency at an acceptable level. To this end, we use an extended version of the Papoulis-Gerchberg algorithm to achieve super-resolution on 3D range data by estimating the unmeasured samples in the environment. In addition, we propose a probabilistic obstacle reconstruction algorithm to consider the probabilities of the estimated (virtual) points and to obtain a very fast prediction about the existence and shape of the obstacles.