ADAMS机器人

Computer Aided Dynamic Simulation of Six-Legged RobotAbhijit Mahapatra1, and Shibendu Shekhar Roy21 Virtual Prototyping & Immersive Visualization Laboratory,Central Mechanical Engineering Research Institute , Durgapur 713209, (CSIR), India.Email: abhi_mahapatra@yahoo.co.in2 Department of Mechanical Engineering, National Institute of Technology, Durgapur 713209, India.Email: ssroy99@Abstract—In this study, kinematics and dynamic simulation of a six-legged robot is performed based on virtual prototyping technology. Modeling and simulation of six-legged robot with three joint legs are carried out with CATIA solid modeler, SimDesigner and ADAMS multi-body dynamic solver. Tripod periodic gait pattern was generated varying the cycle time as well as stroke of the swing of the legs. Variation of joint torques and aggregate center of mass of the robot during periodic gait were analysed. The simulation result provides theoretical basis for developing algorithm for robot motion control.Index Terms—Legged robot, simulation, dynamic modeling, torque, tripod gaitI.I NTRODUCTIONMulti-legged robot locomotion has been an area of keen interest to the researchers over the years because of the advantages of the superior mobility in irregular terrain and the less hazardous influences on environment comparing with the wheeled robots. On the other hand, the requirements for leg coordination and control impose difficulties beyond those encountered in wheeled robots [1]. The dynamic properties of a six-legged robot can significantly influence control response. Such systems are characterized by complex dynamic interactions between individual members connected by actuated joints, including trunk body and relatively less weight legs. Moreover, a six-legged robot in motion is at any one moment a complex combination of open and closed kinematic chains. Legs contacting the ground form closed loops with the trunk body and ground. Non-contacting legs represent individual, branching open kinematic chains. Several algorithms exist for legged robot open and closed chain simulation. The algorithm of Rodriguez and Kreutz [2] uses linear operator methods to derive simple forms of dynamic equations. Another method, that of Lilly and Orin [3] treats a legged robot as a system of multiple manipulators (i.e. legs) contacting an object (i.e. trunk body), with ground contact modeled as a manipulator joint. However, both the models are approximate model of the complex legged robot. In order to design efficient control algorithm for six-legged robot, it is very much essential to develop more accurate dynamic model of real legged robots. In this connection, work of Song and Waldron [1], Shih et al. [4], Pfeiffer et al. [5], Lin, and Song [6], and Kimura et al. [7] are important to mention. Barreto et al. [8] developed the free body diagram method for kinematic and dynamic modeling of a six-legged machine. Erden [9] investigated the dynamics of a hexapod walking robot in a level tripod gait based on Newton-Euler formulation. Koo and Yoon [10] obtained a mathematical model for quadruped walking robot to investigate the dynamics after considering all the inertial effects in the system.Most of the previous researcher developed analytical dynamic model using number of approximation in mechanical structure of legs as well as foot/ground interaction model. Therefore, developed dynamic model was far from actual dynamic behavior.In the present work, an attempt has been made to develop the more accurate dynamic model of the six legged robot using Lagrange-Euler formulation as well as simulate the Computer Aided Design (CAD) model of the robot in ADAMS (Automatic Dynamic Analysis of Mechanical Systems) solver using the concept of rigid multibody dynamics and analyze its dynamic performance.II.M ODELING OF THE S IX-L EGGED R OBOTA three dimensional CAD model [11] of six-legged robot has been developed in Computer Aided Three Dimensional Interactive Application (CATIA) as shown in Fig. 1. The robot trunk body is 495 mm in length, 205 mm in width and 89 mm in height and made of aluminium alloy. The legs are identical and symmetrically distributed around the body on both sides. Each leg consists of three links, namely link 1(tibia), link2 (femur) and link3 (coxa) with effective lengths 83.5mm, 119mm, 98mm respectively. All the joints are motorized revolute joint with rotational axis configuration Z-X-X for the three joints respectively. Total number of DOF of the system is 24 (6 DOF of the trunk body and 18 DOF of the legs). The robot consists of 19 main parts along with 18 servomotors. The three main steps followed to develop virtual prototype of the robot are given below.Step 1 : Part modeling of the different parts of the robot in CATIA V5Step 2: Assembling of the parts in CATIA V5Step 3: Defining of joints and contacts for the robot in CATIA SimDesigner so as to export it to ADAMS for dynamic simulation.The kinematic and dynamic parameters of the robot obtained from CAD model are listed in Table I. Total mass of the six legged robot (including additional parts) = 3.08746 Kg. III. R IGID M ULTIBODY D YNAMIC S IMULATION OF THE R OBOT In order to develop more accurate dynamic model,computer aided simulation tools based on rigid multibodydynamics called Automatic Dynamic Analysis ofMechanical Systems (ADAMS) has been used. A virtualprototype of the six-legged robot has been developed andsimulated in ADAMS/Solver [12]. The followingassumptions are made about the tripod gait to simplifythe rigid multibody dynamic analysis of robot.i. Walk is steadyii. Symmetric, regular and periodic gaits movingforward on the flat terrain.iii. Swing legs are considered not to cross the supportinglegs so that forthcoming support polygon be convexiv. During motion of the robot, three legs are on theground and three legs are in air at every instant,corresponding to 0.5β=.In the present work, following two aspects with respectto dynamic stability of tripod gait over perfectly flat terrain has been checked.a. Variation of torque with stroke at each stepb. Variation of torque with cycle time at each step1z θ link1 (Tibia) link2(Femur)link3 (Coxa)Leg6 Leg4 Leg2 Leg1 Leg3 Leg5 Direction of movement Z Y Figure 1. 3D CAD model of six-legged robot (a) (b) (c) Figure 2. Step Functions for stroke = 0.10m, Cycle time=1.6s for the tripod gait (a) 11L (b) 12L (c) 13LMotions for each of the revolute joints are defined through STEP MATH FUNCTION. As an example, the motion of link 1 having stroke=0.10m and cycle time =1.6s is as shown in Fig. 2. For all type of simulation the values of the joint angles for link 2 and link 3 are taken as 00234&90xxi i θθ==.Impact based contact parameters have been definedbetween the legs and the ground to make the simulationmore realistic (Table II).The simulated robot is evaluated for 60 time steps. Inthe simulation, each time step represents an integrationstep in which the new positions, orientations, velocitiesand acceleration of the robots body parts are computedbased on the forces acting on them.IV. R ESULTS AND D ISCUSSIONSimulation of the six legged robot has been done to investigate the generation of statically stable gait patterns for various stroke and cycle time for the robot. In this section the results of the aggregate center of mass (CM) and torque-distribution in different joint of the links are discussed. A. Variable Aggregate CM of the robotIt is very essential to determine the position of the aggregate CM at any instant of time during motion of the robot to check its dynamic stability. The variation of aggregate CM can be well investigated in ADAMS. For that user written sub-routine was run. Firstly, the user written subroutine centroid.cmd file was imported to the ADAMS workbench and simulation was carried out. Finally, to visualize the variation of the aggregate CM at any instant of time, the user sub-routine AGG_CM_STATE_VARIABLE_IMPORT.cmd was imported. Fig. 3 shows that there is no such off bit path variation of the CM and the trace is more or less a straight line. The system is dynamically stable.B. Joint Torques for Tripod Gait Walking on Flat TerrainThe objective of dynamic torque simulation of the hexapod is minimizing the torque requirements in the leg joints to make the drive more easily controllable. The torques at all joints is calculated by varying the gait parameters. The torque required at the joints, to achieve system motion are simulated in ADAMS environment based on the virtual model in Fig. 1 and dynamic torque equations as mentioned in APPENDIX A. The tripod gait is composed of the following sequences: i) legs 1-4-5 protracting, legs 2-3-6 retracting; ii) legs 2-3-6 protracting, legs 1-3-5 retracting. The walk is repeated for torque-distribution calculations. Fig. 4 shows the torque distribution in the joints of all the links (Duty factor is 0.5, Stroke=0.1m, Cycle time 1.6 secs). In Fig. 4, one can easily observe that for all the legs, the joint torques in retraction take significant values, while the joint torques in protraction are very less values. Again the torque in joint 1 seems to be higher than the torque in other joints. Since these are considered to be proportional to the average dissipated power on the motors, the motor selection has to be based on the maximum torque obtained from the simulation.TABLE II. C ONTACT P ARAMETERS`Figure 3. Aggregate CM variation of hexapod during simulationZXYV. C ONCLUSIONSA 3D virtual prototype robot system has been created CATIA V5 and exported to ADAMS workbench through CATIA SimDesigner to simulate it in real time. The variation of aggregate CG of the system as well as torques at different joints has been studied. The results show that the torque required during retraction phase is much higher than during protraction phase for all the joints. It is obvious that during retraction phase legs are carrying the weight of the trunk body, pay load. Most of the cases torque in joint 1 is higher than the other joints. So motor has to be selected based on the maximum torque. This workwill allow further benchmarking of the mechanical event simulation of the six legged robot system such as trajectory planning, kinematic workspace constraints and coordination issues with other systems references etc. Future work will focus on the relation of energy efficiency with different gait patterns for different walking speeds.A PPENDIX A M ATHEMATICAL F ORMULATION OFD YNAMIC M ODEL The Lagrange’s equation of motion for unconstrained system in a matrix-vector form is()(,)+=M q q h q q Q &&&nc (1)where 2424×∈M R is the mass matrix of the roboticsystem, 24∈h Ris the force vector containing velocity dependent forces and gravitational forces, 24∈Q R nc is the nonconservative force/torque vector applied to the robot.For the constrained six legged robot system due to the foot contact with the ground, one may use Lagrange multipliers and write in matrix-vector form,()(,)λT nc +−=q M q q h q q ΦQ &&& (2) where λ is the vector of Lagrange multipliers whichis identical to the terrain reaction forces of the supporting feet , and 324s n ×∈q ΦR is the constraint Jacobian matrix, s n is the feet grounded.The acceleration constraint equation can be written as(,)−=q Φq γq q 0&&& (3)where 3s n ∈γR is the velocity dependentacceleration vector containing such as the centripetal and coriolis accelerations. The constrained equations of motion of the whole system can be written in vector-matrix form as follows:T nc ⎡⎤−⎧⎫⎧⎫=⎢⎥⎨⎬⎨⎬−⎢⎥⎩⎭⎩⎭⎣⎦q q M Φq Q h Φ0λγ&& (4) The colliding phenomena will occur when the footstrikes the ground. Assuming that the ideally plastic impact occurs between the colliding foot and the ground surface instantaneously, the collision has been modeled as the occurrence of instantaneous velocity change. Here the ideally plastic impact means that the tip velocity of the colliding foot is vanished right after the foot strike. Then from Lagrange’s impulse equation for a kinematically constrained system with the assumption of ideally plastic input, a differential-algebraic equation is obtained in the following form:Figure 4. Variation of joint torques for tripod gait walking on flat terrain (Duty factor is 0.5, Stroke=0.1m, Cycle time 1.6 secs).s T c T q q s s qc c c q q 000000⎡⎤∇⎧⎫⎧⎫⎢⎥⎪⎪⎪⎪⎢⎥λ=⎨⎬⎨⎬⎢⎥⎪⎪⎪⎪λ−⎢⎥⎩⎭⎩⎭⎣⎦M ΦΦΦv Φ& (5)where the vector c 3n c ∈v R is the velocities of the n c colliding feet, s 3n 18s q ×∈ΦR and c 3n 18c q ×∈ΦR are theJacobian matrices of the originally contacted feet and the colliding feet, respectively, s 3n s λ∈R and c 3n c λ∈R are the impact force vectors of the originally contacted feet and the colliding feet, respectively. The instantaneous change of velocity due to the collision offeet is determined as qq q +−=+∆&&& where the superscripts – and + represent quantities right before and after the collision, respectively.A CKNOWLEDGMENTThe authors would like express their sincere thanks to Prof. D.K.Pratihar, IIT, Kharagpur.R EFERENCES[1] S.M. Song, and K. Waldron, Machines that walk: TheAdaptive Suspension Vehicle , MIT Press, Cambridge, MA, 1989.[2] G. Rodriguez, and K. 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