工程科学与技术   2019, Vol. 51 Issue (5): 157-164

Kinematics Solving Method of a Tensegrity Parallel Mechanism Under Actuated
ZHU Wei, LI Hanbing, WANG Chuanwei, GU Kairong
College of Mechanical Eng., Changzhou Univ., Changzhou 213164, China
Abstract: Tensegrity mechanisms are new type mechanism consist of a set of compressive components (rigid rods) and continuous tensile components (ties). The benefits of tensegrity mechanisms in comparison with conventional ones include low mass, small inertia force, good flexibility and foldable. According to configuration principle of tensegrity structure, a 4–SPS type tensegrity mechanism under actuated was designed, which consist of a top platform, a base one and 4 elastic chains (springs) and 4 rigid driven chains (prismatic actuators). The vector equations were deduced from position relation of the mechanism, and the static equilibrium equation also was obtained include the quality of the rigid components. Due to the existence of unconstrained degree of freedom of the mechanism under actuated, analysis of this type of mechanism was more challenging and complicated in comparison with conventional mechanism. Positional positive and negative solution were analyzed through simultaneous minimum potential energy equation and position equation, and the numerical solutions of which also were acquired by using vriable step search method. Finally, velocity and acceleration equations of the mechanism were deduced under the condition of equilibrium, also, motion graphs are drew according to the equations.
Key words: tensegrity    parallel mechanism    kinematic    static equilibrium

1 机构设计及位置分析

4–SPS张拉整体机构的初始平衡位置与4棱柱型张拉整体结构十分相似，如图1所示。图1（a）为4棱柱张拉整体结构，由12根受拉弹性绳索 ${b_i}{b_{i + }}_1$ ${b_{i + }}_1{p_i}$ ${p_i}{p_{i +1 }}$ （细线）和4根受压的刚性杆 $b_{i} p_{i}$ （粗线）组成（ $i=1, 2,$ $3,4$ ，当 $i=4$ 时， $i+1\to1$ ）。根据张拉整体结构要求[16]，为了使结构中节点 $p_i$ 始终保持静力平衡，所受的力必须位于同一平面内，如图1（a）中节点 $p_{1}$ 的受力均位于 ${p_1}{p_3}{b_1}{b_2}$ 平面内。一般地，对于空间棱柱型张拉整体结构[16]，初始平衡位置时上平台相对下平台发生的偏角满足：

 图1 机构原理图 Fig. 1 Schematic diagram of mechanism

 $\alpha = \frac{{\text{π}} }{2} + \frac{{\text{π}} }{\varepsilon }$ (1)

 ${{C}} = \left[ {\begin{array}{*{20}{c}} {{\rm{cos}}\;\;\psi \;\;{\rm{cos}}\;\;\theta\;\;}&{{\rm{cos}}\;\;\psi\;\;{\rm{sin}}\;\;\theta\;\; {\rm{sin}}\;\;\phi - {\rm{sin}}\;\;\psi \;\;{\rm{cos}}\;\;\phi \;\;}&{{\rm{sin}}\;\;\psi\;\; {\rm{sin}}\;\;\phi + {\rm{cos}}\;\;\psi\;\; {\rm{sin}}\;\;\theta\;\; {\rm{cos}}\;\;\phi \;\;} \\ {{\rm{sin}}\;\;\psi \;\;{\rm{cos}}\;\;\theta \;\;}&{{\rm{cos}}\;\;\psi \;\;{\rm{cos}}\;\;\phi + {\rm{sin}}\;\;\psi\;\; {\rm{sin}}\;\;\theta \;\;{\rm{sin}}\;\;\phi \;\;}&{{\rm{sin}}\;\;\psi \;\;{\rm{sin}}\;\;\theta \;\;{\rm{cos}}\;\;\phi - {\rm{cos}}\;\;\psi \;\;{\rm{sin}}\;\;\phi } \\ { - {\rm{sin}}\;\;\theta \;\;}&{{\rm{cos}}\;\;\theta\;\; {\rm{sin}}\;\;\phi \;\;}&{{\rm{cos}}\;\;\theta\;\;{\rm{cos}}\;\;\phi } \end{array}} \right]$ (2)

 \begin{aligned}[b] &{{{r}}_{{b_1}}} = {\left[ {\frac{{\sqrt 2 }}{2}b,0,0} \right]^{\rm{T}}},\;\;\;\;{{{r}}_{{b_2}}} = {\left[ {0,\frac{{\sqrt 2 }}{2}b,0} \right]^{\rm{T}}},\\ &{{{r}}_{{b_3}}} = {\left[ { - \frac{{\sqrt 2 }}{2}b,0,0} \right]^{\rm{T}}},\;\;\;\;{{{r}}_{{b_4}}} = {\left[ {0, - \frac{{\sqrt 2 }}{2}b,0} \right]^{\rm{T}}} \end{aligned} (3)

 \begin{aligned}[b] &{{{u}}_{{p_1}}} = {\left[ { - \frac{a}{2},\frac{a}{2},{\rm{0}}} \right]^{\rm{T}}},\;\;\;\;{{{u}}_{{p_2}}} = {\left[ { - \frac{a}{2}, - \frac{a}{2},0} \right]^{\rm{T}}},\\ &{{{u}}_{{p_3}}} = {\left[ {\frac{a}{2}, - \frac{a}{2},0} \right]^{\rm{T}}},\;\;\;\;{{{u}}_{{p_4}}} = {\left[ {\frac{a}{2},\frac{a}{2},0} \right]^{\rm{T}}} \end{aligned} (4)

 ${{{r}}\!_{{p_i}}} = {{{r}}\!_P} + {{C}}{{{u}}_{{p_i}}},\;\;i=1,2,3,4$ (5)

 ${{{L}}_{{p_i}}} = {{{L}}_{p_{i}}}{{{s}}_{p_{i}}} = {{{r}}_P} + {{C}}{{{u}}_{{p_i}}} - {{{r}}_{{b_i}}}$ (6)

 ${{{L}}_{s_{i}}} = {{{L}}_{s_{i}}}{{{s}}_{s_{i}}} = {{{r}}_P} + {{C}}{{{u}}_{{p_i}}} - {{{r}}_{{b_{i + 1}}}}$ (7)

 ${{{s}}_{p_{i}}} = \frac{{{{{r}}_P} + {{C}}{{{u}}_{{p_i}}} - {{{r}}_{{b_i}}}}}{{{L_{p_{i}}}}}$ (8)
 ${{{s}}_{{s_i}}} = \frac{{{{{r}}_P} + {{C}}{{{u}}_{{p_i}}} - {{{r}}_{{b_{i + 1}}}}}}{{{L_{{s_i}}}}}$ (9)

 图2 刚性支链坐标图 Fig. 2 Rigid chain coordinate diagram

 ${{r}}_{{c_2}}^i = {{{r}}_{{b_i}}} + {l_2}{{{s}}_{{p_i}}}$ (10)
 ${{r}}_{{c_1}}^i = {{{r}}_{{b_i}}} + \left( {{L_{{p_i}}} - {l_1}} \right){{{s}}_{{p_i}}}$ (11)
2 静态平衡分析

 ${U_{\rm s}} = \frac{1}{2}\sum\limits_{i = 1}^4 {{k_i}\delta _i^2}$ (13)

 ${U_{\rm m}} = Mg{{{r}}_P} \cdot {{e}} + \sum\limits_{i = 1}^4 {{m_{{c_1}}}g{{r}}_{{c_1}}^i \cdot {{e}}} + \sum\limits_{i = 1}^4 {{m_{{c_2}}}g{{r}}_{{c_2}}^i \cdot {{e}}}$ (14)

 $U = {U_{\rm s}} + {U_{\rm m}}$ (15)

 $\frac{{\partial U}}{{\partial {{\chi}}}} = 0$ (16)
3 运动学分析 3.1 运动逆解

 $\frac{{\partial U}}{{\partial {{\varTheta}}}} = 0$ (17)

 ${L_{{p_i}}} = \sqrt {{{{L}}^{\rm{T}}_{{p_i}}}{{{L}}_{{p_i}}}}$ (18)
 ${L_{{s_i}}} = \sqrt {{{{L}}^{\rm{T}}_{{s_i}}}{{{L}}_{{s_i}}}}$ (19)

 ${ \left\{\!\!\!\! \begin{array}{l} {L_{{p_1}}} \!\!\!=\!\!\! \left[ \left( \dfrac{b}{2}\left( {\cos\;\phi \sin\;\psi \!-\!\cos\;\psi \sin\;\theta \sin\;\phi \! +\! \cos\;\psi \cos\;\theta } \right)\! -\! \right.\right.\\ \!\! \!\! \left.\left.x \!+\! \dfrac{{\sqrt 2 }}{2}a\! \right)^2 \!+ \! \left( y\! +\! \dfrac{b}{2}(\cos\;\psi \cos\;\phi \! +\! \sin\;\psi \sin\;\theta \sin\;\phi -\right.\right.\\ \left.\left. \!\cos\;\theta \sin\;\psi ) \right)^2 \!+{\left( {{\textit{z}} \!+\! \dfrac{b}{2}\sin\;\theta + \dfrac{b}{2}\cos\;\theta {\sin\;}\phi } \right)^2}\right]^{\tfrac12} ,\\ {L_{{p_2}}}\!\!\! =\!\!\! \left[ \left( x\! +\! \dfrac{b}{2}\left( \cos\;\phi \sin\;\psi \!-\!\cos\;\psi \sin\;\theta \sin\;\phi \!-\!\right.\right.\right.\\ \!\! \left.\left.\left.\cos\;\psi \cos\;\theta \right) \right)^2\! +\left( \!\!\dfrac{b}{2}(\cos\;\psi \cos\;\phi \! +\! \sin\;\psi \sin\;\theta \sin\;\phi \!+ \!\!\right.\right. \\ \left.\left. \cos\;\theta \sin\;\psi ) \!+\!\dfrac{{\sqrt 2 }}{2}a \!\!-\!\! y \!\right)^2\!+{\left( {{\textit{z}} + \dfrac{b}{2}\left( {\sin\;\theta - \cos\;\theta \sin\;\phi } \right)} \right)^2}\right]^{\tfrac 12},\\ {L_{p_3}} \!\!\!= \!\!\left[\! \left( x \!\!+\!\! \dfrac{b}{2}\left( {\cos\;\phi \sin\;\psi \!-\!\cos\;\psi \sin\;\theta \sin\;\phi \! +\! \cos\;\psi \cos\;\theta } \right)\! +\!\right.\right.\\ \!\! \!\!\left.\left. \dfrac{{\sqrt 2 }}{2}a \right)^2 \!+\! \left( y \!-\! \dfrac{b}{2}(\cos\;\psi \cos\;\phi \!+\! \sin\;\psi \sin\;\theta \sin\;\phi \!+\! \right.\right.\\ \left.\left.\cos\;\theta \sin\;\psi ) \right)^2\! +{\left( {\dfrac{b}{2}\left( {\sin\;\theta + \cos\;\theta \sin\;\phi } \right) - {\textit{z}}} \right)^2}\right]^{\tfrac 12},\\ {L_{{p_4}}} \!\!=\!\! \left[ \left( x \!-\! \dfrac{b}{2}\left( \cos\;\phi \sin\;\psi \!-\!\cos\;\psi \sin\;\theta \sin\;\phi \!-\!\right.\right.\right.\\ \left.\left.\left.\cos\;\psi \cos\;\theta \right) \right)^2 \!+\left( y \!+\! \dfrac{b}{2}(\cos\;\psi \cos\;\phi \! +\! \sin\;\psi \sin\;\theta \sin\;\phi \!+\! \right.\right.\\ \left.\left. \cos\;\theta \sin\;\psi ) \!+\! \dfrac{{\sqrt 2 }}{2}a \right)^2+ {\left( {{\textit{z}} + \dfrac{b}{2}\left( {\cos\;\theta \sin\;\phi {{ - \sin\;}}\theta } \right)} \right)^2}\right]^{\tfrac 12} \end{array} \right.}$ (20)
3.2 运动正解

 ${\xi _i}= {{ L}^2_{{p_i}}} - {{{L}}^{\rm{T}}_{{p_i}}} {{{L}}_{{p_i}}}$ (21)

 $V = U + \sum\limits_{i = 1}^4 {{\lambda _i}{\xi _i}}$ (22)

 $\frac{{\partial V}}{{\partial {{\rho}}}} = 0$ (23)

 $\left[ \!\!\!{\begin{array}{*{20}{c}} {{\lambda _1}} \\ {{\lambda _2}} \\ {{\lambda _3}} \\ {{\lambda _4}} \end{array}} \!\!\!\right] = - {\left[ \!\!\!{\begin{array}{*{20}{c}} {\displaystyle\frac{{\partial {\xi _1}}}{{\partial x}}}&{\displaystyle\frac{{\partial {\xi _2}}}{{\partial x}}}&{\displaystyle\frac{{\partial {\xi _3}}}{{\partial x}}}&{\displaystyle\frac{{\partial {\xi _4}}}{{\partial x}}} \\ {\displaystyle\frac{{\partial {\xi _1}}}{{\partial y}}}&{\displaystyle\frac{{\partial {\xi _2}}}{{\partial y}}}&{\displaystyle\frac{{\partial {\xi _3}}}{{\partial y}}}&{\displaystyle\frac{{\partial {\xi _4}}}{{\partial y}}} \\ {\displaystyle\frac{{\partial {\xi _1}}}{{\partial {\textit{z}}}}}&{\displaystyle\frac{{\partial {\xi _2}}}{{\partial {\textit{z}}}}}&{\displaystyle\frac{{\partial {\xi _3}}}{{\partial {\textit{z}}}}}&{\displaystyle\frac{{\partial {\xi _4}}}{{\partial {\textit{z}}}}} \\ {\displaystyle\frac{{\partial {\xi _1}}}{{\partial \psi }}}&{\displaystyle\frac{{\partial {\xi _2}}}{{\partial \psi }}}&{\displaystyle\frac{{\partial {\xi _3}}}{{\partial \psi }}}&{\displaystyle\frac{{\partial {\xi _4}}}{{\partial \psi }}} \end{array}}\!\!\! \right]^{ - 1}}\left[\!\!\! {\begin{array}{*{20}{c}} {\displaystyle\frac{{\partial U}}{{\partial x}}} \\ {\displaystyle\frac{{\partial U}}{{\partial y}}} \\ {\displaystyle\frac{{\partial U}}{{\partial {\textit{z}}}}} \\ {\displaystyle\frac{{\partial U}}{{\partial \psi }}} \end{array}} \!\!\!\right]$ (24)

4 速度和加速度分析 4.1 动平台

 \left\{ \begin{aligned} & {\omega _x} = {c_{13}}{{\dot c}_{12}} + {c_{23}}{{\dot c}_{22}} + {c_{33}}{{\dot c}_{32}} , \\ & {\omega _y} = {c_{11}}{{\dot c}_{13}} + {c_{21}}{{\dot c}_{23}} + {c_{31}}{{\dot c}_{33}}, \\ & {\omega _x} = {c_{12}}{{\dot c}_{11}} + {c_{22}}{{\dot c}_{21}} + {c_{32}}{{\dot c}_{31}} \\ \end{aligned} \right. (25)

 \left\{ \begin{aligned} & {\omega _x} = - \dot \psi \sin\;\theta + \dot \phi , \\ & {\omega _y} = \dot \psi \cos\;\theta \sin\;\phi + \dot \theta \cos\;\phi , \\ & {\omega _{\textit{z}}} = \dot \psi \cos\;\theta \cos\;\phi - \dot \theta \sin\;\phi \\ \end{aligned} \right. (26)

 ${{\omega}} = {{{K}}_1}{\dot{{\varTheta }}}$ (27)

 ${\dot{{\omega}}} = {{{K}}_1}{\ddot{{\varTheta}}} + {{{K}}_2}{\dot{{\varTheta }}}$ (28)

 \begin{aligned}[b] {{{K}}_2} = \left[\!\!\! {\begin{array}{*{20}{c}} { - \displaystyle\frac{1}{2}\dot \theta \cos\;\theta }&\!\!\!{ - \displaystyle\frac{1}{2}\dot \psi \cos\;\theta }&\!\!\!0 \\ { - \displaystyle\frac{1}{2}\dot \theta \sin\;\theta \sin\;\phi + \frac{1}{2}\dot \phi \cos\;\theta \cos\;\phi }&\!\!\!{ - \displaystyle\frac{1}{2}\dot \phi \sin\;\theta \sin\;\phi - \frac{1}{2}\dot \phi \sin\;\phi } &\!\!\!{\displaystyle\frac{1}{2}\dot \psi \cos\;\theta \cos\;\phi - \frac{1}{2}\dot \theta \sin\;\phi } \\ { - \displaystyle\frac{1}{2}\dot \theta \sin\;\theta \cos\;\phi + \frac{1}{2}\dot \phi \cos\;\theta \sin\;\phi }&\!\!\!{ - \displaystyle\frac{1}{2}\dot \phi \sin\;\theta \cos\;\phi - \frac{1}{2}\dot \phi \cos\;\phi }&\!\!\!{\displaystyle\frac{1}{2}\dot \psi \cos\;\theta \sin\;\phi - \frac{1}{2}\dot \theta \cos\;\phi } \end{array}} \!\!\!\right] {\text{。}} \end{aligned}
4.2 驱动支链

 ${{\dot{{r}}}_{{p_i}}} = {{\dot{{r}}}_P} + {{\omega}} \times {{{u}}_{{p_i}}}$ (29)

 ${{\ddot{{r}}}_{{p_i}}} = {{\ddot{{r}}}_P} + {\dot{{\omega}}} \times {{{u}}_{{p_i}}} + {{\omega}} \times \left( {{{\omega}} \times {{{u}}_{{p_i}}}} \right)$ (30)

 ${{{r}}_{{p_i}}} = {{{r}}_{{b_i}}} + {{ L}_i}{{{s}}_i}$ (31)

 ${{\dot{{r}}}_{{p_i}}} = {\dot { L}_i}{{{s}}_i} + {{ L}_i}\left( {{{\dot{{\varOmega}}}_i} \times {{{s}}_i}} \right)$ (32)

 ${{{s}}_{{p_i}}} \times {{\dot{{r}}}_{{p_i}}} = {{ L}_{{p_i}}}{{{s}}_{{p_i}}} \times \left( {{{\dot{{\varOmega}}}_i} \times {{{s}}_i}} \right)= {{ L}_{{p_i}}}{{\dot{{\varOmega}}}_i}$ (33)

 ${{\ddot{{\varOmega}}}_i} = \frac{1}{{{{ L}_{{p_i}}}}}\left( {{{{s}}_{{p_i}}} \times {{\ddot{{r}}}}_{{p_i}}} + \left( {{{\dot{{\varOmega}}}_i} \times {{{s}}_{{p_i}}}} \right) \times {{{{\dot{{r}}}}_{{p_i}}}} \right) - \frac{{{{\dot { L}}_{{p_i}}}}}{{{ L}_{{p_i}}^2}}{{{s}}_{{p_i}}} \times {{\dot{{r}}}_{{p_i}}}$ (34)

 ${{\ddot{{\varOmega}}}_i} = \frac{1}{{{{ L}_i}}}{{{s}}_i} \times \left[ {{{{\ddot{{r}}}}_P} + {\dot{{\omega}}} \times {{{u}}_{{p_i}}}} \right]$ (35)
5 数值算例

 图3 样机模型 Fig. 3 Prototype model

5.1 运动正反解计算

 图4 机构运动位姿结构 Fig. 4 Configuration of the mechanism for the kinematic solution

5.2 逆运动仿真

 图5 各支链长度变化 Fig. 5 Length of each chain

 图6 驱动支链角速度 Fig. 6 Angular velocity of drive chain

 图7 驱动支链角加速度 Fig. 7 Angular acceleration of drive chain

5.3 样机验证

 图8 平衡位置 Fig. 8 Equilibrium position

6 结　论

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