Dynamic Gait Programming Of Biped Robot Based On Moving Horizon Optimization

Biped walking robot can cooperate with human beings in humanized life and working environment without special large-scale modification of the environment, so biped walking robot gets wide potentials in application. With the continuous development of exploration of space such as moon and the Mars, biped robot with its advantages of flexibility can play an important role in landing part. Additionally, as active prosthetics, biped robot can provide walking tool for lower limb disabled persons in both indoor and outdoor environments.Biped robot has been recognized as research hotspot for several decades; theoretic researches in different aspects are verified through numerous simulations and actual applications. Researchers have done a lot work from simplest planar mechanical model to the most advanced humanoid robot made by HONDA and SONY. Research topics include designing, modeling, dynamic analysis and gait synthesis of walking robot.However distance to actual application in real world is still far. Most traditional bipedal robots utilized control strategy composed of offline programming and online tracking, whereas condition of road surface is complex. In order to make stable walking of biped robot in diverse conditions, real-time gait programming research is required. Besides, high speed development of computer technology makes online gait programming realizable. This paper focuses on real-time gait programming problem of planar biped walking robot.First of all, kinematic and kinetic mathematical models of planar seven-link biped robot are established. In kinematics, binary tree data structure is used to describe links'characteristics and mutual connections. A kinematics simulation toolbox for MATLAB is established. In dynamics, identical continuous dynamic model of both single support phase(SSP) and double support phase(DSP) and discrete dynamical model at impact of swing foot landing are established. Landing models of toe and heel are only different at relative contact constraints corresponding to Lagrange items. Hybrid system model including three continuous phases and three discrete phases are composed, and further reduced unified dimensional hybrid zero dynamic model is developed by partial feedback linearization.Then a kind of modified nonlinear model predictive control(NMPC) realtime gait programming strategy based on moving optimization idea is proposed for real-time gait programming problem of biped robot. Through setting kinetic and kinematic virtual constraints to SSP and three subphases of DSP, complex realtime gait programming problem is transformed into four NMPC problems using quadratics of generalized inputs in predictive horizon as cost functions. Simulation results showed that this method could help to implement realtime gait programming of BIP robot realizing dynamic walking with foot rotation, and biped stability is satisfied without sliding. So effectiveness and applicability of the approach are verified.Third, stable walking is the most important one among all control tasks. This paper analyze robust periodic stability of stable multi-step walking under model uncertainty and alteration of ground condition for gait trajectory of anthropomorphic biped robot generated by moving optimization realtime gait programming method. It is proved that periodic stability of multi-step and single-step is consistent. By offline integration on time axis, simulation was carried out to analyze the influences of model uncertainties and velocity jump errors to the existence of limit cycle and contractive rate.Fourth, real-time gait programming of biped robot based on moving optimization can be formalized as NMPC problems subjected to piecewise state constraints. Considering that some certain constraints must be satisfied strictly in actual robot control and in order to improve computational efficiency, an improved fast numerical algorithm(state transformation method) is proposed based on implicit state constraint method. Considering piecewise characteristic of effective domain of state constraint in predictive horizon, modular and smoothing method are utilized to make resulted piecewise state constraints share the same canonical form as cost function and is continuous and differentiable such that first order derivatives of cost function and state constraints with respect to control parameters can be derived by the same Hamiltonian method.Finally, pure real-time gait programming method of biped robot based on moving optimization needs to solve dynamical optimization problem including state constraints online at each sampling point which requires complex computational process and heavy computational load. Besides, realtime gait programming method based on moving optimization defined fixed predictive horizon which did not cover the whole gait period. Quite a few virtual kinematic and kinetic constraints should be defined to guarantee humanized stable gait trajectories of biped robot. So this paper proposes a real-time control strategy whose principle is the composition of both moving horizon optimization and virtual passive dynamic walking. Differential characteristic of mechanical energy of seven link robot under virtual gravity force is analyzed, furthermore, input torques in joints constrained to this energy law are designed and determined. Through optimized regulation of links'virtual inclination, expected velocities and state parameters of system at the end point of gait period, differences between robot states and expected values at the very beginning of next period just after landing of swing foot is minimized. Compared with gait programming method applying only moving horizon optimization, this approach needs less optimization parameters and the length of moving horizon is variable which guarantee the feasibility and optimality of robot gait at the beginning of next period. Simulation analysis showed that this approach realized dynamic walking motion including foot rotation and initial points of multi-step walking satisfied periodic stability requirement under identical ground condition.