【MR】几何雅可比vs解析雅可比 + python加载Indy7的URDF逆动力学仿真(MR+Pybullet)对比...

Python加载urdf解析为MR参数,Pybullet加载urdf,逆动力学仿真

知识点:

几何雅可比,就是关节空间速度映射到末端的空间速度

解析雅可比,就是不同姿态表示下所求的导数,比如欧拉角表示法,用来表示末端姿态的变化,通过末端欧拉角速度向角速度的转换,需要通过一个雅可比矩阵来完成

解析雅可比与几何雅可比关系的推导:

【MR】几何雅可比vs解析雅可比 + python加载Indy7的URDF逆动力学仿真(MR+Pybullet)对比..._第1张图片

【MR】几何雅可比vs解析雅可比 + python加载Indy7的URDF逆动力学仿真(MR+Pybullet)对比..._第2张图片

【MR】几何雅可比vs解析雅可比 + python加载Indy7的URDF逆动力学仿真(MR+Pybullet)对比..._第3张图片解析雅可比计算推导

Python加载URDF解析为MR参数:(含解析雅可比计算)

from urdfpy import URDF
import numpy as np
import modern_robotics as mr


def TransformToXYZ(T):
  return np.array(T[0:3,3])


def getJoint(robot,link_name):
  ret_link=[]
  for link in robot.links:
    if link.name == link_name:
      ret_link=link
      break;
  return ret_link


def w_p_to_Slist(w,p,ROBOT_DOF):
    Slist = []
    for i in range(0,ROBOT_DOF):
      w_ = w[i];
      p_ = p[i];      
      v_ = -np.cross(w_,p_)
      Slist.append([w_[0],w_[1],w_[2],v_[0],v_[1],v_[2]])
    return np.transpose(Slist)


#几何雅可比转解析雅可比-后增加
def analyticaljacobian(M,Blist, thetalist):
    Jb =  mr.JacobianBody(Blist, thetalist)
    Tsb = mr.FKinBody(M,Blist, thetalist)
    Ajacobian = np.vstack(
        (np.hstack((np.identity(3), np.zeros((3, 3)))),
         np.hstack((np.zeros((3, 3)), np.linalg.inv(Tsb[0:3,0:3]))))
        ).dot(Jb)
    return  Ajacobian
#原版本
def AnalyticJacobianBody(M,Blist, thetalist):
    """Computes the Analytic Jacobian for an open chain robot


    :param Blist: The joint screw axes in the end-effector frame when the
                  manipulator is at the home position, in the format of a
                  matrix with axes as the columns
    :param thetalist: A list of joint coordinates
    :return: The body Jacobian corresponding to the inputs (6xn real
             numbers)


    Example Input:
        Blist = np.array([[0, 0, 1,   0, 0.2, 0.2],
                          [1, 0, 0,   2,   0,   3],
                          [0, 1, 0,   0,   2,   1],
                          [1, 0, 0, 0.2, 0.3, 0.4]]).T
        thetalist = np.array([0.2, 1.1, 0.1, 1.2])
    Output:
        np.array([[-0.04528405, 0.99500417,           0,   1]
                  [ 0.74359313, 0.09304865,  0.36235775,   0]
                  [-0.66709716, 0.03617541, -0.93203909,   0]
                  [ 2.32586047,    1.66809,  0.56410831, 0.2]
                  [-1.44321167, 2.94561275,  1.43306521, 0.3]
                  [-2.06639565, 1.82881722, -1.58868628, 0.4]])
    """
    Jb =  mr.JacobianBody(Blist, thetalist)
    Tsb = mr.FKinBody(M,Blist, thetalist)
    Rsb= Tsb[0:3,0:3]
    #r = mr.so3ToVec(Rsb)  #SO3  这里有问题 应为 ?mr.so3ToVec(MatrixLog3(Rsb))
    #norm_r = np.sqrt(r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
    #omega_r = mr.VecToso3(r) #转轴矢量的so3
    #A = np.eye(3) - (1-np.cos(norm_r))/(norm_r*norm_r) * omega_r+ (norm_r - np.sin(norm_r)) / (norm_r**3) * omega_r @ omega_r
    A_ = np.eye(6)
    A_[0:3,0:3] = Rsb
    A_[3:6,3:6] = Rsb
    Ja = A_ @ Jb
    return Ja


def loadURDF(urdf_name):
  print(urdf_name)
  robot = URDF.load(urdf_name)
  LinkNum = len(robot.links)
  JointNum = len(robot.actuated_joints)


  p_ = []  
  w_= []
  M_list = [np.eye(4)]


  Glist =[]
  com_p_=[]
  for joint in robot.actuated_joints:
    child_link = getJoint(robot,joint.child)
    p_.append(TransformToXYZ(robot.link_fk()[child_link]))
    G = np.eye(6)
    G[0:3,0:3] = child_link.inertial.inertia
    G[3:6,3:6] = child_link.inertial.mass*np.eye(3)
    Glist.append(G)


    child_M = robot.link_fk()[child_link] 
    child_M_R, child_M_p =mr.TransToRp(child_M)
    child_w = np.array(child_M_R @ np.array(joint.axis).T)
    w_.append( child_w ) 
    child_M[0:3,0:3] = np.eye(3)
    CoM_M =  child_M@ child_link.inertial.origin


    M_list.append(CoM_M)




  eef_link = getJoint(robot,robot.end_links[0].name)#获取末端连杆的关节
  M = robot.link_fk()[eef_link]  


  M_list.append(M)
  Slist = w_p_to_Slist(w_,p_,JointNum)  
  Blist = mr.Adjoint(mr.TransInv(M))@ Slist
  Mlist = []
  for i in range(1,len(M_list)):
    M_i_1 = M_list[i-1]
    M_i = M_list[i]  
    Mlist.append(mr.TransInv(M_i_1) @ M_i)
  Mlist = np.array(Mlist)  
  Glist = np.array(Glist)


  return {"M":M,"Slist" : Slist,"Blist": Blist,"Mlist":Mlist,"Glist":Glist,"actuated_joints_num":len(robot.actuated_joints)}

indy7_sim.p(Pybullet与MR逆动力学仿真对比)

import pybullet as p
import time
import pybullet_data
import numpy as np
np.set_printoptions(precision=6, suppress=True)
import math
from modern_robotics import *
import modern_robotics as mr
import numpy as np
from mr_urdf_loader import loadURDF
from mr_urdf_loader import *


def pos_orn_to_T(pos,orn):
  T = np.eye(4)
  T[0:3,3] = np.array(pos).T
  T[0:3,0:3] = np.reshape(p.getMatrixFromQuaternion(orn),(3,3))
  return T


#几何雅可比转解析雅可比  与
def analyticaljacobian(M,Blist, thetalist):
    Jb =  mr.JacobianBody(Blist, thetalist)
    Tsb = mr.FKinBody(M,Blist, thetalist)
    Ajacobian = np.vstack(
        (np.hstack((np.identity(3), np.zeros((3, 3)))),
         np.hstack((np.zeros((3, 3)), np.linalg.inv(Tsb[0:3,0:3]))))
        ).dot(Jb)
    return  Ajacobian


urdf_name = "V:/learn/MR/download/mr_urdf_loader-main/example/indy7/indy7/indy7.urdf"


## Modern Robotics setup
MR=loadURDF(urdf_name)
M  = MR["M"]
Slist  = MR["Slist"]
Mlist  = MR["Mlist"]
Glist  = MR["Glist"]
Blist  = MR["Blist"]
actuated_joints_num = MR["actuated_joints_num"]
## pybullet setup
p.connect(p.GUI)
p.setGravity(0, 0, -9.8)
p.setAdditionalSearchPath(pybullet_data.getDataPath())


robotID = p.loadURDF(urdf_name, [0, 0, 0],[0, 0, 0, 1],useFixedBase=1)
numJoints = p.getNumJoints(robotID)
p.resetBasePositionAndOrientation(robotID, [0, 0, 0], [0, 0, 0, 1])
for i in range(0,numJoints):
  p.setJointMotorControl2(robotID, i, p.VELOCITY_CONTROL, targetVelocity=0, force=0)


#p.resetJointState(robotID,2,1) 无明显变化


useRealTimeSim = False
p.setRealTimeSimulation(useRealTimeSim)
timeStep = 1/240.0;
p.setTimeStep(timeStep)


q = [0,0,0,0,0,0] #q = [0,0,0,0,0,0]
q_dot = [0,0,0,0,0,0]#[0,0,0,0,0,0]
q_ddot = [0,1,0,0,0,0]
g = np.array([0,0,-9.8])
Ftip = [0,0,0,0,0,0]


print(Glist[0][0:3,0:3])
print(Glist[1][0:3,0:3]) 
print(Glist[2][0:3,0:3]) 
print(Glist[3][0:3,0:3]) 
''' print(Glist[4][0:3,0:3]) 
print(Glist[5][0:3,0:3])  '''


print(p.getDynamicsInfo(robotID, 1)[2])  #主惯性矩
print(p.getDynamicsInfo(robotID, 2)[2]) 
print(p.getDynamicsInfo(robotID, 3)[2]) 
print(p.getDynamicsInfo(robotID, 4)[2]) 


while p.isConnected():
  # get Joint Staes
  jointStates = np.array(p.getJointStates(robotID,[1,2,3,4,5,6]),dtype=object)
  # get Link Staes  
  linkState = np.array(p.getLinkState(robotID,7,1,1),dtype=object)
  q = np.array(jointStates[:,0])
  q_dot = np.array(jointStates[:,1])


  # Pybullet Forward Kinematics 
  pb_Tsb = pos_orn_to_T(linkState[0],linkState[1])
  # Modern Robotics Forward Kinematics   
  mr_Tsb = mr.FKinSpace(M,Slist,q)
  mr_Tsb = mr.FKinBody(M,Blist,q)




  # Pybullet Jacobian
  #pb_J =  p.calculateJacobian(robotID,4,[0,0,0],[q[0],q[1],q[2]],[q_dot[0],q_dot[1],q_dot[2]],[0,0,0])
  # Modern Robotics Jacobian  
  mr_Jb= JacobianBody(Blist, q)
  mr_Js= JacobianSpace(Slist, q)  
  #mr_ja  mr_ja2 结果一样
  mr_Ja = AnalyticJacobianBody(M,Blist, q)  # pb_j = mr_Ja  
  mr_Ja2=analyticaljacobian(M,Blist, q)
  #print("=============mr_Ja=============") 
  #print(mr_Ja)
  #print("=============mr_Ja2=============")
  #print(mr_Ja2)
  #pybullet InverseDynamics    
  print(q)
  pb_ID= np.array(p.calculateInverseDynamics(robotID,[q[0],q[1],q[2],q[3],q[4],q[5] ],[q_dot[0],q_dot[1],q_dot[2],q_dot[3],q_dot[4],q_dot[5] ] ,[0,0,0,0,0,0] ))
  #modern_robotics InverseDynamics
  mr_ID =mr.InverseDynamics(q, q_dot, q_ddot, g, Ftip, Mlist,  Glist, Slist)


  print("=============pb_Tsb=============")
  print(pb_Tsb)
  print("=============mr_Tsb=============")
  print(mr_Tsb)
  print("=============pb_ID=============")
  print(pb_ID)
  print("=============mr_ID=============")
  print(mr_ID)


  #set torques
  for i in range(0,actuated_joints_num):    
    p.setJointMotorControl2(robotID, i+1, p.TORQUE_CONTROL,force=mr_ID[i])
    #p.setJointMotorControl2(robotID, i+1, p.TORQUE_CONTROL,force=pb_ID[i])


  p.stepSimulation()
  time.sleep(timeStep)

indy7_urdf_loader.py测试加载urdf解析为MR参数程序

import numpy as np
np.set_printoptions(precision=6, suppress=True)
from mr_urdf_loader import loadURDF
from modern_robotics import *
urdf_name = "V:/learn/MR/download/mr_urdf_loader-main/example/indy7/indy7/indy7.urdf"
MR=loadURDF(urdf_name)
M  = np.array(MR["M"])
Slist  = np.array(MR["Slist"])
Mlist  = np.array(MR["Mlist"])
Glist  = np.array(MR["Glist"])
Blist  = np.array(MR["Blist"])
actuated_joints_num = MR["actuated_joints_num"]


thetalist = np.array([0,0,np.pi/2.0])
dthetalist = np.array([0,0,0.1])
ddthetalist = np.array([0,0,0])
g = np.array([0,0,-9.8])
Ftip = [0,0,0,0,0,0]




#print(M)
print(Mlist)
print(Glist)
print("FKinSpace\n", FKinSpace(M,Slist,thetalist))
print("FKinBody\n", FKinBody(M,Blist,thetalist))


print("JacobianSpace\n", JacobianSpace(Slist,thetalist))
print("JacobianBody\n", JacobianBody(Blist,thetalist))


print("InverseDynamics\n" ,InverseDynamics(thetalist, dthetalist, ddthetalist, g, Ftip, Mlist,  Glist, Slist))

测试加载URDF解析为MR参数程序的输出:

PS V:\learn\MR\download\mr_urdf_loader-main>  v:; cd 'v:\learn\MR\download\mr_urdf_loader-main'; & 'C:\Users\cxy\AppData\Local\Programs\Python\Python39\python.exe' 'c:\Users\cxy\.vscode\extensions\ms-python.python-2022.16.1\pythonFiles\lib\python\debugpy\adapter/../..\debugpy\launcher' '63759' '--' 'v:\learn\MR\download\mr_urdf_loader-main\example\indy7\indy7_urdf_loader.py' 
V:/learn/MR/download/mr_urdf_loader-main/example/indy7/indy7/indy7.urdf
[[[ 1.      0.      0.      0.    ]
  [ 0.      1.      0.      0.    ]
  [ 0.      0.      1.      0.0775]
  [ 0.      0.      0.      1.    ]]


 [[ 1.      0.      0.      0.    ]
  [ 0.      1.      0.     -0.109 ]
  [ 0.      0.      1.      0.222 ]
  [ 0.      0.      0.      1.    ]]


 [[ 1.      0.      0.     -0.    ]
  [ 0.      1.      0.      0.0305]
  [ 0.      0.      1.      0.45  ]
  [ 0.      0.      0.      1.    ]]


 [[ 1.      0.      0.     -0.    ]
  [ 0.      1.      0.      0.075 ]
  [ 0.      0.      1.      0.267 ]
  [ 0.      0.      0.      1.    ]]


 [[ 1.      0.      0.      0.    ]
  [ 0.      1.      0.     -0.114 ]
  [ 0.      0.      1.      0.083 ]
  [ 0.      0.      0.      1.    ]]


 [[ 1.      0.      0.     -0.    ]
  [ 0.      1.      0.     -0.069 ]
  [ 0.      0.      1.      0.168 ]
  [ 0.      0.      0.      1.    ]]


 [[ 1.     -0.      0.      0.    ]
  [ 0.      1.     -0.     -0.    ]
  [ 0.      0.      1.      0.06  ]
  [ 0.      0.      0.      1.    ]]]
[[[ 0.154186 -0.000002  0.000017  0.        0.        0.      ]
  [-0.000002  0.12937  -0.048543  0.        0.        0.      ]
  [ 0.000017 -0.048543  0.059644  0.        0.        0.      ]
  [ 0.        0.        0.       11.80301   0.        0.      ]
  [ 0.        0.        0.        0.       11.80301   0.      ]
  [ 0.        0.        0.        0.        0.       11.80301 ]]


 [[ 0.29357  -0.        0.000014  0.        0.        0.      ]
  [-0.        0.280941  0.03728   0.        0.        0.      ]
  [ 0.000014  0.03728   0.036206  0.        0.        0.      ]
  [ 0.        0.        0.        7.992921  0.        0.      ]
  [ 0.        0.        0.        0.        7.992921  0.      ]
  [ 0.        0.        0.        0.        0.        7.992921]]


 [[ 0.034246  0.000001  0.000007  0.        0.        0.      ]
  [ 0.000001  0.03406   0.00186   0.        0.        0.      ]
  [ 0.000007  0.00186   0.004505  0.        0.        0.      ]
  [ 0.        0.        0.        2.991341  0.        0.      ]
  [ 0.        0.        0.        0.        2.991341  0.      ]
  [ 0.        0.        0.        0.        0.        2.991341]]


 [[ 0.006704  0.000004  0.000002  0.        0.        0.      ]
  [ 0.000004  0.002792 -0.00128   0.        0.        0.      ]
  [ 0.000002 -0.00128   0.006193  0.        0.        0.      ]
  [ 0.        0.        0.        2.12317   0.        0.      ]
  [ 0.        0.        0.        0.        2.12317   0.      ]
  [ 0.        0.        0.        0.        0.        2.12317 ]]


 [[ 0.009949  0.        0.000003  0.        0.        0.      ]
  [ 0.        0.009782 -0.000935  0.        0.        0.      ]
  [ 0.000003 -0.000935  0.002715  0.        0.        0.      ]
  [ 0.        0.        0.        2.288651  0.        0.      ]
  [ 0.        0.        0.        0.        2.288651  0.      ]
  [ 0.        0.        0.        0.        0.        2.288651]]


 [[ 0.000435  0.       -0.        0.        0.        0.      ]
  [ 0.        0.000445  0.000001  0.        0.        0.      ]
  [-0.        0.000001  0.000596  0.        0.        0.      ]
  [ 0.        0.        0.        0.400839  0.        0.      ]
  [ 0.        0.        0.        0.        0.400839  0.      ]
  [ 0.        0.        0.        0.        0.        0.400839]]]
FKinSpace
 [[ 0.     -0.     -1.     -0.578 ]
 [ 0.      1.     -0.     -0.1865]
 [ 1.     -0.      0.      0.7495]
 [ 0.      0.      0.      1.    ]]
FKinBody
 [[ 0.     -0.     -1.     -0.578 ]
 [ 0.      1.     -0.     -0.1865]
 [ 1.     -0.      0.      0.7495]
 [ 0.      0.      0.      1.    ]]
JacobianSpace
 [[ 0.      0.      0.      0.      0.      0.    ]
 [ 0.     -1.     -1.     -0.     -1.     -0.    ]
 [ 1.      0.      0.      1.     -0.      1.    ]
 [-0.      0.2995  0.7495 -0.0035  1.0995 -0.1865]
 [-0.      0.      0.      0.      0.      0.    ]
 [-0.      0.      0.      0.      0.      0.    ]]
JacobianBody
 [[ 1.     -0.     -0.      0.     -0.      0.    ]
 [-0.     -1.     -1.      0.     -1.      0.    ]
 [ 0.      0.      0.      1.      0.      1.    ]
 [-0.     -0.578  -0.578   0.183  -0.228   0.    ]
 [-0.578   0.      0.     -0.     -0.     -0.    ]
 [-0.1865  0.45   -0.      0.     -0.     -0.    ]]
InverseDynamics
 [ 0.000019 -0.        0.      ]

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