To effectively identify structural modal parameters during multi-joint combined self-excitation motion of a robot, this paper proposes dynamic spatial sensitivity analysis and the torque projection matrix. By utilizing dynamic spatial sensitivity analysis, we can determine the operating range of joint angles during self-excited motion. Additionally, based on the torque projection matrix, we can ascertain the motion direction of each joint in the multi-joint combined movement, ensuring that the energy and direction of the excitation signal meet the white noise requirements.
Dynamic sensitivity coefficient
In order to evaluate the influence of the angular displacement of the joint on the dynamic characteristics of the robot, the dynamic sensitivity coefficient εi of the i order mode is defined as the ratio of the change of natural frequency to the original natural frequency.
$${\varepsilon _i}\left( {a,b} \right)=\frac{{\left| {{\omega _{ib}} – {\omega _{ia}}} \right|}}{{{\omega _{ia}}}},i=1,2, \cdots ,n$$
(1)
where ωi.a. represents the natural frequency of the ith order mode before the change of robot motion state; ωib represents the natural frequency of the ith order mode after the change of motion state; n represents the maximum modal order of the system in the frequency band under consideration. The dynamic equation of robot structure is as follows
$$M\ddot {x}+C\dot {x}+Kx=F(t)$$
(2)
where M represents the mass matrix of the structure; C represents the structural damping matrix; K represents the stiffness matrix; x represents the displacement vector of structural vibration; \(F\left( t \right)\) represents the external stimuli to the structure. The characteristic equation of the structure
$${\omega _i}^{2}M{\left\{ \phi \right\}_i}=K{\left\{ \phi \right\}_i}$$
(3)
where ωi represents natural frequency of the ith mode of the structure; \({\left\{ \phi \right\}_i}\) represents the ith mode of the structure; When the mass mr is added to any r degree of freedom of the structure, the dynamic equation of the structure
$$\left( {M+\Delta m} \right)\ddot {x}+C\dot {x}+Kx=F\left( t \right)$$
(4)
where \(\Delta m\)represents the additional mass is introduced into the mass matrix
$$\Delta m=\left[ {\begin{array}{*{20}{c}} 0&{}&{}&{}&{} \\ {}& \ddots &{}&{}&{} \\ {}&{}&{{m_r}}&{}&{} \\ {}&{}&{}& \ddots &{} \\ {}&{}&{}&{}&0 \end{array}} \right]$$
(5)
The characteristic equation of the structure after adding mass
$${\omega _i}{^{\prime 2}}(M+\Delta m){\left\{ \phi \right\}_i}^{\prime }=K{\left\{ \phi \right\}_i}^{\prime }$$
(6)
where ωi represents natural frequency of the first mode after adding mass to the structure; \({\left\{ \phi \right\}_i}^{\prime }\) represents the ith mode of the structure after adding mass; When the additional mass is small enough, the mode of the structure before and after the additional mass is basically unchanged
$${\left\{ \phi \right\}_i} \approx {\left\{ \phi \right\}_i}^{\prime }$$
(7)
The eigenequation (3) before and after the additional mass is subtracted from Eq. (6), and the Eq. (7) is substituted
$$\omega _{i} ^{2} M\left\{ \phi \right\}_{i} – \omega ^{\prime2}\left( {M + \Delta m} \right)\left\{ \phi \right\}_{i} = 0$$
(8)
When \({\left\{ \phi \right\}_i}\) is the mass normalized mode vector of the structure, the above formula can be further expressed as
$${\omega _i}{^{\prime2}}\left( {{{\left\{ \phi \right\}}_i}^{H}\Delta m{{\left\{ \phi \right\}}_i}+1} \right)={\omega _i}^{2}$$
(9)
Divide both sides of the above equation by ωi2 and take the root of it
$$\frac{{{\omega _i}^{\prime }}}{{{\omega _i}}}=\sqrt {\frac{1}{{1+{m_r}{{\left| {{\phi _{ir}}} \right|}^2}}}}$$
(10)
where \({\phi _{ir}}\) represents the mode component of the ith mode vector of the structure at the position of r degree of freedom. By bringing the above formula into Eq. (1), the modal expression of the dynamic sensitivity coefficient of the ith mode can be obtained
$${\varepsilon _i}\left( r \right)=\sqrt {\frac{1}{{1+{\alpha _i}^{2}{m_r}{{\left| {{\varphi _{ir}}} \right|}^2}}}} – 1$$
(11)
where \({\varphi _{ir}}\) represents the ith normalized modal component of the structure at the position of r degree of freedom.
According to Eq. (10), the dynamic sensitivity coefficient is proportional to the modal shapes of the structure. The larger the structure’s modal shapes, the greater the change of natural frequency due to variation in joint angles; The smaller the modal shapes of the structure, the less the change of natural frequency caused by joint angle degree variation. The validity of this analysis method will be verified experimentally in Sect. 3.
Torque projection matrix
The forces and torques applied on the end-effector of robot can be represented as \(\:F={\left(Fx,Fy,Fz,mx,my,mz\right)}^{T}\). The forces and torques of each joint of robot can be combined as the joint vector \(\:\tau\:={\left({\tau\:}_{1},{\tau\:}_{2},{\tau\:}_{3},{\tau\:}_{4},{\tau\:}_{5},{\tau\:}_{6}\right)}^{T}\). When the driving torques produced from the rotation of the joints cause an output at the end-effector, the sum of the virtual work at each joint is given by:
$$\:W={\tau\:}^{T}\delta\:q={\tau\:}_{1}\delta\:{q}_{1}+{\tau\:}_{2}\delta\:{q}_{2}+\cdots\:{+\tau\:}_{6}\delta\:{q}_{6}$$
(12)
Where \(\:\delta\:q\) represents the virtual displacement in the joint space. The virtual work of the end-effector is given by:
$$\:W={F}^{T}D={f}_{x}dx+{f}_{y}dy+{f}_{z}dz+{m}_{x}{\delta\:}_{x}+{m}_{y}{\delta\:}_{y}+{m}_{z}{\delta\:}_{z}$$
(13)
where \(\:D\) represents the virtual displacement in the workspace. Since there is a geometric relationship between the forces acting on the end-effector and the joint displacements, it is assumed that the virtual displacement in the workspace is \(\:\varvec{D}=\varvec{J}\varvec{\delta\:}\varvec{q}\), where \(\:\varvec{J}\) represents the Jacobian matrix. According to the principle of virtual work, the virtual work produced by the change in the joint vector is equal to the virtual work in the workspace. Therefore, we have:
$$\:{\tau\:}^{T}\delta\:q={F}^{T}D={F}^{T}J\delta\:q$$
(14)
Simplify:
$$\:\tau\:={J}^{T}F$$
(15)
Equation (15) represents the mapping relationship between the generalized external forces at the robot’s end-effector and the joint torques can be expressed as. The velocity Jacobian matrix can be written as:
$$\:\dot{\varvec{x}}=J\left(q\right)\dot{q}$$
(16)
Where \(\:\dot{\varvec{q}}\) represents the joint velocity vector and \(\:\dot{\varvec{x}}\) represents the operational velocity vector. When the robot undergoes self-excited motion based on joint motors, the forces acting on the robot system are generated by the driving torques of each joint. At this point, the input forces \(\:\left\{{f}_{1},{\:f}_{2},\:\cdots\:,\:{f}_{d}\right\}\) combined form the input torques vector \(\:\varvec{f}\) which is equivalent to the joint torque vector \(\:\tau\:={\left({\tau\:}_{1},{\tau\:}_{2},{\tau\:}_{3},{\tau\:}_{4},{\tau\:}_{5},{\tau\:}_{6}\right)}^{T}\) in Eq. (13). Therefore, by combining Eq. (15), we obtain:
$$\:D\left(\tau\:\right)={\left[\begin{array}{ccc}\begin{array}{c}{\tau\:}_{1x}\\\:{\tau\:}_{2x}\\\:\begin{array}{c}\vdots\\\:{\tau\:}_{6x}\end{array}\end{array}&\:\begin{array}{c}{\tau\:}_{1y}\\\:{\tau\:}_{2y}\\\:\begin{array}{c}\vdots\\\:{\tau\:}_{6y}\end{array}\end{array}&\:\begin{array}{c}{\tau\:}_{1z}\\\:{\tau\:}_{2z}\\\:\begin{array}{c}\vdots\\\:{\tau\:}_{6z}\end{array}\end{array}\end{array}\right]}_{6\times\:3}$$
(17)
Where \(\:D\left(\tau\:\right)\) represents the projection of the joint torque vector in any coordinate system along the three axes; \(\:{\tau\:}_{6x}\) represents the projection of the input torque of the sixth joint along the x-axis direction; \(\:{\tau\:}_{6y}\) represents the projection of the input torque of the sixth joint along the y-axis direction; \(\:{\tau\:}_{6z}\) represents the projection of the input torque of the sixth joint along the z-axis direction. Equation (17) is a non-square matrix, and its rank is equal to the number of the largest linearly independent groups. Therefore, when the joint torque has projection components along the x, y, and z axes in any cartesian coordinate system, \(\:D\left(\tau\:\right)\) is a full-rank matrix.
link