TRAC-IK机器人运动学求解器

　　TRAC-IK和Orocos KDL类似，也是一种基于数值解的机器人运动学求解器，但是在算法层面上进行了很多改进(Specifically, KDL’s convergence algorithms are based on Newton’s method, which does not work well in the presence of joint limits — common for many robotic platforms. TRAC-IK concurrently runs two IK implementations. One is a simple extension to KDL’s Newton-based convergence algorithm that detects and mitigates local minima due to joint limits by random jumps. The second is an SQP (Sequential Quadratic Programming) nonlinear optimization approach which uses quasi-Newton methods that better handle joint limits. By default, the IK search returns immediately when either of these algorithms converges to an answer )，相比KDL求解效率（成功率和计算时间）高了很多。

　　下面在Ubuntu16.04中安装TRAC-IK（之前已经安装过ROS Kinetic）：

sudo apt-get install ros-kinetic-trac-ik

　　按照ROS教程新建一个名为ik_test的Package，并创建urdf文件夹用于存放机器人URDF描述文件，创建launch文件夹存放launch文件：

　　参考trac_ik_examples修改package.xml以及CMakeLists.txt文件，添加TRAC-IK以及KDL的支持。编写一个简单的robot.urdf文件，joint1为与基座link0相连的基关节，joint3为末端关节：

<robot name="test_robot">
    <link name="link0" />
    <link name="link1" />
    <link name="link2" />
    <link name="link3" />

    <joint name="joint1" type="continuous">
        <parent link="link0"/>
        <child link="link1"/>
        <origin xyz="0 0 0" rpy="0 0 0" />
        <axis xyz="1 0 0" />
    </joint>

    <joint name="joint2" type="continuous">
        <parent link="link1"/>
        <child link="link2"/>
        <origin xyz="0 0 1" rpy="0 0 0" />
        <axis xyz="1 0 0" />
    </joint>

    <joint name="joint3" type="continuous">
        <parent link="link2"/>
        <child link="link3"/>
        <origin xyz="0 0 1" rpy="0 0 0" />
        <axis xyz="1 0 0" />
    </joint>

</robot>

　　TRAC-IK求解机器人逆运动学函数为CartToJnt：

int CartToJnt(const KDL::JntArray &q_init, const KDL::Frame &p_in, KDL::JntArray &q_out, const KDL::Twist& bounds=KDL::Twist::Zero());

　　第一个参数q_init为关节的初始值，p_in为输入的末端Frame，q_out为求解输出的关节值。基本用法如下：

#include <trac_ik/trac_ik.hpp>

TRAC_IK::TRAC_IK ik_solver(KDL::Chain chain, KDL::JntArray lower_joint_limits, KDL::JntArray upper_joint_limits, double timeout_in_secs=0.005, double error=1e-, TRAC_IK::SolveType type=TRAC_IK::Speed);  

% OR

TRAC_IK::TRAC_IK ik_solver(string base_link, string tip_link, string URDF_param="/robot_description", double timeout_in_secs=0.005, double error=1e-, TRAC_IK::SolveType type=TRAC_IK::Speed);  

% NOTE: The last arguments to the constructors are optional.
% The type can be one of the following:
% Speed: returns very quickly the first solution found
% Distance: runs for the full timeout_in_secs, then returns the solution that minimizes SSE from the seed
% Manip1: runs for full timeout, returns solution that maximizes sqrt(det(J*J^T))
% Manip2: runs for full timeout, returns solution that minimizes cond(J) = |J|*|J^-1|

int rc = ik_solver.CartToJnt(KDL::JntArray joint_seed, KDL::Frame desired_end_effector_pose, KDL::JntArray& return_joints, KDL::Twist tolerances);

% NOTE: CartToJnt succeeded if rc >=0   

% NOTE: tolerances on the end effector pose are optional, and if not
% provided, then by default are 0.  If given, the ABS() of the
% values will be used to set tolerances at -tol..0..+tol for each of
% the 6 Cartesian dimensions of the end effector pose.

　　下面是一个简单的测试程序，先通过KDL计算正解，然后使用TRAC-IK反算逆解：

#include "ros/ros.h"
#include <trac_ik/trac_ik.hpp>

#include <kdl/chainiksolverpos_nr_jl.hpp>
#include <kdl/chain.hpp>
#include <kdl/chainfksolver.hpp>
#include <kdl/chainfksolverpos_recursive.hpp>
#include <kdl/frames_io.hpp>

using namespace KDL;

int main(int argc, char **argv)
{
    ros::init(argc, argv, "ik_test");
    ros::NodeHandle nh("~");

    int num_samples;
    std::string chain_start, chain_end, urdf_param;
    double timeout;
    const double error = 1e-;

    nh.param("chain_start", chain_start, std::string(""));
    nh.param("chain_end", chain_end, std::string(""));

    if (chain_start=="" || chain_end=="") {
        ROS_FATAL("Missing chain info in launch file");
        exit (-);
    }

    nh.param("timeout", timeout, 0.005);
    nh.param("urdf_param", urdf_param, std::string("/robot_description"));

    if (num_samples < )
        num_samples = ;

    TRAC_IK::TRAC_IK ik_solver(chain_start, chain_end, urdf_param, timeout, error, TRAC_IK::Speed);  

    KDL::Chain chain;
    bool valid = ik_solver.getKDLChain(chain);

    if (!valid) {
        ROS_ERROR("There was no valid KDL chain found");
        return -;
    }

    // Set up KDL IK
    KDL::ChainFkSolverPos_recursive fk_solver(chain); // Forward kin. solver based on kinematic chain

    // Create joint array
    unsigned int nj = chain.getNrOfJoints();
    ROS_INFO ("Using %d joints",nj);
    KDL::JntArray jointpositions = JntArray(nj);

   // Assign some values to the joint positions
    for(unsigned int i=;i<nj;i++){
        float myinput;
        printf ("Enter the position of joint %i: ",i);
        scanf ("%e",&myinput);
        jointpositions(i)=(double)myinput;
    }

    // Create the frame that will contain the results
    KDL::Frame cartpos;    

    // Calculate forward position kinematics
    bool kinematics_status;
    kinematics_status = fk_solver.JntToCart(jointpositions,cartpos);

    Vector p = cartpos.p;   // Origin of the Frame
    Rotation M = cartpos.M; // Orientation of the Frame

    double roll, pitch, yaw;
    M.GetRPY(roll,pitch,yaw);

    if(kinematics_status>=){
        printf("%s \n","KDL FK Succes");
        std::cout <<"Origin: " << p() << "," << p() << "," << p() << std::endl;
        std::cout <<"RPY: " << roll << "," << pitch << "," << yaw << std::endl;

    }else{
        printf("%s \n","Error: could not calculate forward kinematics :(");
    }

    KDL::JntArray joint_seed(nj);
    KDL::SetToZero(joint_seed);
    KDL::JntArray result(joint_seed);

    int rc=ik_solver.CartToJnt(joint_seed,cartpos,result);
    if(rc < )
        printf("%s \n","Error: could not calculate forward kinematics :(");
    else{
        printf("%s \n","TRAC IK Succes");
        for(unsigned int i = ; i < nj; i++)
            std::cout << result(i) << " ";
    }

    return ;
}

　　test.launch文件如下：　　

<?xml version="1.0"?>
<launch>
  <arg name="chain_start" default="link0" />
  <arg name="chain_end"   default="link3" />
  <arg name="timeout" default="0.005" />

  <param name="robot_description" textfile="$(find ik_test)/urdf/robot.urdf" />

  <node name="ik_test" pkg="ik_test" type="ik_test" output="screen">
    <param name="chain_start" value="$(arg chain_start)"/>
    <param name="chain_end" value="$(arg chain_end)"/>
    <param name="timeout" value="$(arg timeout)"/>
    <param name="urdf_param" value="/robot_description"/>
  </node>

</launch>

　　使用catkin_make编译成功，并设置环境后，运行该程序

roslaunch ik_test test.launch 

　　通过键盘分别输入三个关节值：0，1.5708（90°），0  运动学正逆解计算结果如下图所示：

　　

　　接下来使用7自由的的Franka panda机器人进行正逆解计算测试。

　　franka_description中包含Franka机器人的URDF文件，编写panda_test.launch，设置基关节为panda_link0，末端关节（法兰盘）为panda_link8。因此，正逆解计算都是以机器人法兰中心为基准。

<?xml version="1.0"?>
<launch>
  <arg name="chain_start" default="panda_link0" />
  <arg name="chain_end"   default="panda_link8" />
  <arg name="timeout" default="0.005" />

  <param name="robot_description" command="$(find xacro)/xacro.py '$(find franka_description)/robots/panda_arm.urdf.xacro'" />

  <node name="ik_test" pkg="ik_test" type="ik_test" output="screen">
    <param name="chain_start" value="$(arg chain_start)"/>
    <param name="chain_end" value="$(arg chain_end)"/>
    <param name="timeout" value="$(arg timeout)"/>
    <param name="urdf_param" value="/robot_description"/>
  </node>

</launch>

　　test.cpp中计算逆解前使用getKDLLimits函数得到机器人关节运动范围，并设定关节初始值在上下限的正中间。

#include "ros/ros.h"
#include <trac_ik/trac_ik.hpp>

#include <kdl/chainiksolverpos_nr_jl.hpp>
#include <kdl/chain.hpp>
#include <kdl/chainfksolver.hpp>
#include <kdl/chainfksolverpos_recursive.hpp>
#include <kdl/frames_io.hpp>

using namespace KDL;

int main(int argc, char **argv)
{
    ros::init(argc, argv, "ik_test");
    ros::NodeHandle nh("~");

    int num_samples;
    std::string chain_start, chain_end, urdf_param;
    double timeout;
    const double error = 1e-;

    nh.param("chain_start", chain_start, std::string(""));
    nh.param("chain_end", chain_end, std::string(""));

    if (chain_start=="" || chain_end=="") {
        ROS_FATAL("Missing chain info in launch file");
        exit (-);
    }

    nh.param("timeout", timeout, 0.005);
    nh.param("urdf_param", urdf_param, std::string("/robot_description"));

    if (num_samples < )
        num_samples = ;

    TRAC_IK::TRAC_IK ik_solver(chain_start, chain_end, urdf_param, timeout, error, TRAC_IK::Distance);  

    KDL::Chain chain;
    bool valid = ik_solver.getKDLChain(chain);

    if (!valid) {
        ROS_ERROR("There was no valid KDL chain found");
        return -;
    }

    KDL::JntArray ll, ul; //lower joint limits, upper joint limits
    valid = ik_solver.getKDLLimits(ll,ul);

    if (!valid)
        ROS_INFO("There were no valid KDL joint limits found");

    // Set up KDL IK
    KDL::ChainFkSolverPos_recursive fk_solver(chain); // Forward kin. solver based on kinematic chain

    // Create joint array
    unsigned int nj = chain.getNrOfJoints();
    ROS_INFO ("Using %d joints",nj);
    KDL::JntArray jointpositions = JntArray(nj);

   // Assign some values to the joint positions
    for(unsigned int i=;i<nj;i++){
        float myinput;
        printf ("Enter the position of joint %i: ",i);
        scanf ("%e",&myinput);
        jointpositions(i)=(double)myinput;
    }

    // Create the frame that will contain the results
    KDL::Frame cartpos;    

    // Calculate forward position kinematics
    bool kinematics_status;
    kinematics_status = fk_solver.JntToCart(jointpositions,cartpos);

    Vector p = cartpos.p;   // Origin of the Frame
    Rotation M = cartpos.M; // Orientation of the Frame

    double roll, pitch, yaw;
    M.GetRPY(roll,pitch,yaw);

    if(kinematics_status>=){
        printf("%s \n","KDL FK Succes");
        std::cout <<"Origin: " << p() << "," << p() << "," << p() << std::endl;
        std::cout <<"RPY: " << roll << "," << pitch << "," << yaw << std::endl;

    }else{
        printf("%s \n","Error: could not calculate forward kinematics :(");
    }

    KDL::JntArray joint_seed(nj);
    //KDL::SetToZero(joint_seed);

    for (uint j=; j<joint_seed.data.size(); j++)
        joint_seed(j) = (ll(j)+ul(j))/2.0;

    KDL::JntArray result(joint_seed);

    int rc=ik_solver.CartToJnt(joint_seed,cartpos,result);
    if(rc < )
        printf("%s \n","Error: could not calculate inverse kinematics :(");
    else{
        printf("%s \n","TRAC IK Succes");
        for(unsigned int i = ; i < nj; i++)
            std::cout << result(i) << " ";
    }

    return ;
}

　　为了验证TRAC-IK计算结果，使用libfranka中的echo_robot_state先输出机器人当前状态。O_T_EE是末端执行器（这里没有安装末端执行器，因此就是以法兰中心为基准）在机器人基坐标系中的位置姿态矩阵（按列优先存储），q为关节角度。

"O_T_EE": [0.722009,-0.690627,-0.0416898,0,-0.691463,-0.72236,-0.00866381,0,-0.0241316,0.0350823,-0.999093,0,0.371908,0.00215211,0.428567,1],
"O_T_EE_d": [0.722007,-0.690631,-0.041642,0,-0.691467,-0.722355,-0.00871983,0,-0.0240581,0.0350898,-0.999095,0,0.371922,0.00215033,0.428596,1],
"F_T_EE": [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],
"EE_T_K": [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],
"elbow": [-0.0168831,-1],
"elbow_d": [-0.0168889,-1],
"q": [0.0180582,-0.537152,-0.0168831,-2.60095,0.0297504,2.03967,0.753258],
"dq": [-0.00051007,-0.000212418,0.00125443,0.000394549,-2.9871e-05,0.000278746,0.000261954],
...

　　开启测试程序，输入上面的关节值，输出如下：

　　KDL正解计算的值与O_T_EE一致。根据Franka机器人的DH参数表，可以写出从法兰到基座的变换矩阵：

　　将正解输入与逆解输出的关节值分别代入变换矩阵中，可以发现理论计算与O_T_EE和KDL正解结果一致。

　　由于7自由度机械臂有无数组逆解，因此一般要根据某种优化原则，选取其中一组。对于Franka机械臂，可使用TRAC-IK得到逆解后进行关节空间中的轨迹规划，实现类似于“movej”的功能。

参考：

trac_ik

trac_ik_examples

KDL Geometric primitives

API reference of the Kinematics and Dynamics Library

机械臂运动学逆解（Analytical solution）

orocos_kdl学习（一）：坐标系变换

orocos_kdl学习（二）：KDL Tree与机器人运动学

MoveIt!中的运动学插件