CN103009381B - Method for rapidly constructing Jacobi of less-degree-of-freedom parallel mechanism containing special branched chain structure - Google Patents
Method for rapidly constructing Jacobi of less-degree-of-freedom parallel mechanism containing special branched chain structure Download PDFInfo
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Abstract
本发明涉及一种含特殊支链结构的少自由度并联机构的雅可比快速构造方法。本发明方法定义了一系列特殊支链的结构特征，通过计算支链中驱动关节的驱动力螺旋就可以快速构造出少自由度并联机构的部分雅可比矩阵。在计算支链约束力螺旋的基础上，可以获取并联机构的整体雅可比矩阵。这种方法在构造部分雅可比矩阵时，不需要借助于螺旋的互易运算或正交运算，因此显著提高了少自由度并联机构的部分雅可比矩阵的计算效率。再提供支链约束力螺旋，就可以快速地获得少自由度并联机构的整体雅可比矩阵。
The invention relates to a Jacobian rapid construction method of a parallel mechanism with few degrees of freedom containing a special branched chain structure. The method of the invention defines the structural features of a series of special branched chains, and the partial Jacobian matrix of the parallel mechanism with few degrees of freedom can be quickly constructed by calculating the driving force helix of the driving joints in the branched chains. On the basis of calculating the branch-chain binding force helix, the overall Jacobian matrix of the parallel mechanism can be obtained. This method does not need to rely on the reciprocal operation or orthogonal operation of the spiral when constructing the partial Jacobian matrix, so the calculation efficiency of the partial Jacobian matrix of the parallel mechanism with few degrees of freedom is significantly improved. Furthermore, by providing the branched constraint helix, the overall Jacobian matrix of the parallel mechanism with few degrees of freedom can be quickly obtained.
Description
技术领域 technical field
本发明涉及一种少自由度并联机构的雅可比快速构造方法。 The invention relates to a Jacobian rapid construction method of a parallel mechanism with few degrees of freedom.
背景技术 Background technique
少自由度并联机构是指自由度数少于6的空间并联机构；少自由度并联机构的速度、精度或刚度分析比6自由度并联机构更复杂。雅可比矩阵，简称雅可比，用来描述机构中各驱动关节速度与末端执行器速度之间的映射关系，它也是分析并联机构精度与刚度的重要工具。雅可比矩阵的计算效率对少自由度并联机构设计和控制都有重要作用。 A parallel mechanism with few degrees of freedom refers to a space parallel mechanism with less than 6 degrees of freedom; the speed, accuracy or stiffness analysis of a parallel mechanism with few degrees of freedom is more complicated than that of a parallel mechanism with 6 degrees of freedom. The Jacobian matrix, referred to as Jacobian, is used to describe the mapping relationship between the speed of each driving joint in the mechanism and the speed of the end effector. It is also an important tool for analyzing the accuracy and stiffness of parallel mechanisms. The computational efficiency of the Jacobian matrix plays an important role in the design and control of parallel mechanisms with few degrees of freedom.
对于6自由度并联机构的雅可比构造，可以通过将各条支链看作一个串联机构来获得关节速度与末端执行器速度之间的关系，从而得到雅可比矩阵。但少自由度并联机构的雅可比矩阵不能采用这种方法获得，因为少自由度并联机构的支链雅可比矩阵不是方阵，不能求逆。因此，人们采用了一些其它方法来构造少自由度并联机构的雅可比矩阵，例如，Joshi和Tsai于2002年在《ASME Journal of Mechanical Design》（124(6):第254–258页）的论文“Jacobian analysis of limited-DOF parallel manipulators”中利用运动螺旋与约束力螺旋之间的互易运算，获得锁定驱动关节后新增的支链约束螺旋，进而联立各支链的螺旋方程求解少自由度并联机构的雅可比矩阵。Huang于2004年在《Proceedings of the 11th World Congress in Mechanism and Machine Science》（天津会议，4月1日，第65–76页）的论文“The kinematics and type synthesis of lower-mobility parallel robot manipulator”中通过在支链中增加虚拟连杆和虚拟关节，将少自由度并联机构扩充为一个虚拟的6自由度并联机构，采用6自由度并联机构雅可比构造方法来获得少自由度并联机构的雅可比矩阵。Hong和Choi于2011年在《ASME Journal of Mechanisms and Robotics》（3(1):第1–6页）的论文“Formulation of unique form of screw based Jacobian for lower mobility parallel manipulators”中通过少自由度并联机构动平台承载力与关节驱动力之间的关系获得少自由度并联机构的雅可比矩阵，他们的方法中需要计算与驱动关节运动螺旋不正交而与支链其它运动螺旋均正交的力螺旋。这些方法需要借助于螺旋的互易运算、正交运算，或扩展成虚拟的6自由度并联机构，因而计算量很大。如果我们能够充分利用少自由度并联机构可能存在的特殊支链结构，那么雅可比矩阵的计算效率可以大大提高。 For the Jacobian structure of the 6-DOF parallel mechanism, the relationship between the joint velocity and the end-effector velocity can be obtained by considering each branch chain as a series mechanism, so as to obtain the Jacobian matrix. But the Jacobian matrix of the parallel mechanism with few degrees of freedom cannot be obtained by this method, because the branched Jacobian matrix of the parallel mechanism with few degrees of freedom is not a square matrix and cannot be inverted. Therefore, some other methods have been used to construct the Jacobian matrix of parallel mechanisms with few degrees of freedom, for example, Joshi and Tsai in the paper of "ASME Journal of Mechanical Design" (124(6): 254–258) in 2002 "Jacobian analysis of limited-DOF parallel manipulators" uses the reciprocal operation between the motion helix and the constraint helix to obtain the newly added branch chain constraint helix after locking the drive joint, and then solve the helix equation of each branch chain with less freedom The Jacobian matrix of a parallel mechanism. Huang in the paper "The kinematics and type synthesis of lower-mobility parallel robot manipulator" in "Proceedings of the 11th World Congress in Mechanism and Machine Science" (Tianjin Conference, April 1, pp. 65–76) in 2004 By adding virtual connecting rods and virtual joints in the branch chain, the parallel mechanism with few degrees of freedom is expanded into a virtual parallel mechanism with 6 degrees of freedom, and the Jacobian of the parallel mechanism with 6 degrees of freedom is obtained by using the Jacobian construction method of the parallel mechanism with 6 degrees of freedom matrix. Hong and Choi in 2011 in the paper "Formulation of unique form of screw based Jacobian for lower mobility parallel manipulators" in "ASME Journal of Mechanisms and Robotics" (3(1): pp. 1–6) through the parallel manipulators with few degrees of freedom The relationship between the bearing capacity of the dynamic platform of the mechanism and the driving force of the joints is used to obtain the Jacobian matrix of the parallel mechanism with few degrees of freedom. Their method needs to calculate the force that is not orthogonal to the helix of the driving joint motion but is orthogonal to the other helixes of the branch chain spiral. These methods need the reciprocal operation and orthogonal operation of the screw, or be extended to a virtual 6-DOF parallel mechanism, so the calculation is very heavy. If we can make full use of the special branched structure that may exist in the parallel mechanism with few degrees of freedom, then the computational efficiency of the Jacobian matrix can be greatly improved.
发明内容 Contents of the invention
本发明的目的在于提供一种针对含特殊支链结构的少自由度并联机构的雅可比快速构造方法。这种方法不需要借助于螺旋的互易运算、正交运算，也无需将少自由度并联机构扩展成虚拟的6自由度并联机构，因此雅可比矩阵的运算量很小，计算效率很高。 The object of the present invention is to provide a Jacobian rapid construction method for a parallel mechanism with few degrees of freedom containing a special branched chain structure. This method does not need the reciprocal operation and orthogonal operation of the spiral, nor does it need to expand the parallel mechanism with few degrees of freedom into a virtual 6-degrees-of-freedom parallel mechanism. Therefore, the calculation amount of the Jacobian matrix is small and the calculation efficiency is high.
本发明采用的技术方案的步骤如下： The steps of the technical solution adopted in the present invention are as follows:
(1)特殊结构少自由度并联机构的支链类型为下述中任一种： (1) The branch type of parallel mechanism with special structure and few degrees of freedom is any of the following:
(i)若少自由度并联机构的支链中含有1个带驱动器的移动副时，驱动器移动方向应该与支链中其它移动副运动方向垂直，并且驱动器的移动方向应该通过支链中各转动副的轴线，并通过支链中各万向副和球副的中心；具体结构包括但不限于以下5种类型： (i) If the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver, the moving direction of the driver should be perpendicular to the moving direction of the other moving pairs in the branch chain, and the moving direction of the driver should pass through each rotation pair in the branch chain. The axis of the pair, and through the center of each universal pair and spherical pair in the branch chain; the specific structure includes but is not limited to the following 5 types:
(a)少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个球副，且移动副的运动方向通过2个球副的中心； (a) The branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two ball pairs at its two ends, and the motion direction of the moving pair passes through the centers of the two ball pairs;
(b)少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的1个球副以及1个万向副，且移动副的运动方向通过球副的中心和万向副的中心； (b) The branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver, a ball pair and a universal pair at both ends, and the movement direction of the moving pair passes through the center of the ball pair and the universal direction vice center;
(c)少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个万向副，且移动副的运动方向通过2个万向的中心 ； (c) The branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two universal pairs at both ends, and the moving direction of the moving pair passes through the centers of the two universals;
(d)少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的1个万向副和1个球副，且移动副的运动方向通过球副的中心和转动副的轴线； (d) The branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver, a universal joint and a ball pair at both ends, and the movement direction of the moving pair passes through the center of the ball pair and the rotating pair axis;
(e)少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个移动副，且带驱动器的移动副的运动方向垂直于另外2个移动副的运动方向； (e) The branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two moving pairs located at its two ends, and the moving direction of the moving pair with a driver is perpendicular to the moving direction of the other two moving pairs;
(ii)若少自由度并联机构的支链中含有1个带驱动器的转动副时，驱动器的轴线方向应该与支链中其它转动副轴线方向垂直，而支链中移动副的位置和姿态可以任意配置；具体结构包括但不限于以下类型： (ii) If the branch chain of the parallel mechanism with few degrees of freedom contains a rotating pair with a driver, the axis direction of the driver should be perpendicular to the axis direction of other rotating pairs in the branch chain, and the position and attitude of the moving pair in the branch chain can be arbitrary Configuration; specific structures include but are not limited to the following types:
(f)少自由度并联机构的支链中含有1个带驱动器的转动副，带驱动器的转动副一端串联有3个移动副和2个转动副，且带驱动器的转动副的轴线垂直于另外2个转动副的轴线； (f) The branch chain of the parallel mechanism with few degrees of freedom contains a revolving pair with a driver, and one end of the revolving pair with a driver is connected in series with 3 moving pairs and 2 revolving pairs, and the axis of the revolving pair with a driver is perpendicular to the other Axes of 2 revolving pairs;
(2) 计算支链的驱动力螺旋： (2) Calculate the driving force spiral of the branch chain:
对于少自由度并联机构的支链中带有驱动器的移动副，移动副的驱动力螺旋是一个力， $ a pri = τ pri ( f ; r × f )，其中τ pri 为力的大小，|| f ||2=1，矢量 f 的方向与带有驱动器的移动副运动方向一致， r 为带有驱动器的移动副在全局坐标系中的位置矢量；对于支链中带有驱动器的转动副，转动副的驱动力螺旋是一个力偶， $ a rev = τ rev (0; c )，其中τ rev 为力偶的大小，|| c ||2=1，矢量 c 的方向与转动副轴线方向一致； For the moving pair with a driver in the branch chain of the parallel mechanism with few degrees of freedom, the driving force of the moving pair is a force, $ a pri = τ pri ( f ; r × f ), where τ pri is the magnitude of the force,| | f || 2 =1, the direction of the vector f is consistent with the motion direction of the moving pair with the driver, r is the position vector of the moving pair with the driver in the global coordinate system; for the rotating pair with the driver in the branch chain , the driving force spiral of the rotating pair is a force couple, $ a rev = τ rev (0; c ), where τ rev is the size of the force couple, || c || 2 =1, the direction of the vector c is consistent with the direction of the rotating pair axis ;
(3) 构造少自由度并联机构的部分雅可比矩阵：部分雅可比矩阵将少自由度并联机构的动平台的速度和角速度映射为驱动关节运动速度，含特殊支链结构的少自由度并联机构的部分雅可比矩阵 J partial 的快速构造公式如下： (3) Construct the partial Jacobian matrix of the parallel mechanism with few degrees of freedom: the partial Jacobian matrix maps the velocity and angular velocity of the moving platform of the parallel mechanism with few degrees of freedom to the motion velocity of the driving joints, and the parallel mechanism with few degrees of freedom with special branch chain structure The quick construction formula of the partial Jacobian matrix J partial is as follows:
(式1) (Formula 1)
其中，N表示机构自由度即驱动器的数目，且N<6； f i （i = 1, 2, …, N）表示各个驱动力螺旋的力分量， c i （i = 1, 2, …, N）表示各个驱动力螺旋的力偶分量， r E 表示动平台中心到螺旋坐标系原点的矢量； Among them, N represents the degree of freedom of the mechanism, that is, the number of actuators, and N <6; f i ( i = 1, 2, ..., N ) represents the force component of each driving force screw, c i ( i = 1, 2, ..., N ) represents the force couple component of each driving force spiral, r E represents the vector from the center of the moving platform to the origin of the spiral coordinate system;
(4) 计算支链的约束力螺旋：根据支链约束力螺旋对并联机构动平台不做功的原理，计算出6-N个约束力螺旋（N表示机构自由度），记为 $ c j = ( f c j ; c c j )（j = 1, 2, …, 6-N），其中 f c j 是约束力螺旋的力分量， c c j 是约束力螺旋的力偶分量； (4) Calculating the binding force spiral of the branch chain: According to the principle that the branch chain binding force spiral does not do work on the parallel mechanism dynamic platform, calculate 6- N binding force spirals ( N represents the degree of freedom of the mechanism), which is recorded as $ c j = ( f c j ; c c j )( j = 1, 2, …, 6- N ), where f c j is the force component of the constraint force helix, c c j is the force couple component of the constraint force helix;
(5) 构造少自由度并联机构的整体雅可比矩阵：整体雅可比矩阵将少自由度并联机构的动平台所承受的外部载荷映射为关节驱动力和支链约束力，整体雅可比矩阵是一个6行6列的方阵，含特殊支链结构的少自由度并联机构的整体雅可比矩阵 J overall 的快速构造公式如下： (5) Construct the overall Jacobian matrix of the parallel mechanism with few degrees of freedom: the overall Jacobian matrix maps the external load borne by the moving platform of the parallel mechanism with few degrees of freedom to the driving force of the joint and the constraint force of the branch chain. The overall Jacobian matrix is a A square matrix with 6 rows and 6 columns, the quick construction formula of the overall Jacobian matrix J overall of the parallel mechanism with few degrees of freedom and special branched chain structure is as follows:
(式2)。 (Formula 2).
根据步骤(1)分析少自由度并联机构可能存在的特殊支链结构。 According to step (1), the possible special branched structure of the parallel mechanism with few degrees of freedom is analyzed.
根据步骤(2)计算支链的驱动力螺旋。 Calculate the driving force helix of the branched chain according to step (2).
根据步骤(3)和步骤(5)构造少自由度并联机构的部分雅可比矩阵和整体雅可比矩阵。 According to step (3) and step (5), the partial Jacobian matrix and the overall Jacobian matrix of the parallel mechanism with few degrees of freedom are constructed.
本发明具有的有益效果是： The beneficial effects that the present invention has are:
(1) 提出的部分雅可比矩阵构造方法只需要计算驱动力螺旋，不需要借助于螺旋的互易运算或正交运算，显著提高了少自由度并联机构的部分雅可比矩阵的计算效率； (1) The proposed partial Jacobian matrix construction method only needs to calculate the driving force spiral, and does not need to rely on the reciprocal operation or orthogonal operation of the spiral, which significantly improves the calculation efficiency of the partial Jacobian matrix of the parallel mechanism with few degrees of freedom;
(2) 提出的整体雅可比矩阵构造方法不需要将少自由度并联机构扩展为虚拟的6自由度并联机构，减少了整体雅可比矩阵的计算量； (2) The proposed overall Jacobian matrix construction method does not need to expand the parallel mechanism with few degrees of freedom to a virtual 6-degree-of-freedom parallel mechanism, which reduces the calculation amount of the overall Jacobian matrix;
(3) 能够用于构造符合特殊支链结构要求的任何少自由度并联机构的雅可比矩阵； (3) It can be used to construct the Jacobian matrix of any parallel mechanism with few degrees of freedom that meets the requirements of special branched chain structures;
(4) 能够用于构造部分雅可比矩阵和整体雅可比矩阵。 (4) It can be used to construct partial Jacobian matrix and overall Jacobian matrix.
附图说明 Description of drawings
图1是SPS特殊结构支链示意图。 Figure 1 is a schematic diagram of the branched chain of SPS special structure.
图2是UPS特殊结构支链示意图。 Figure 2 is a schematic diagram of UPS special structure branch.
图3是UPU特殊结构支链示意图。 Fig. 3 is a schematic diagram of U P U branched chain with special structure.
图4是RPS特殊结构支链示意图。 Fig. 4 is a schematic diagram of branched chains with special structure of RPS .
图5是PPP特殊结构支链示意图。 Figure 5 is a schematic diagram of the branched chain of the special structure of PPP .
图6是PPPRRR特殊结构支链示意图。 Figure 6 is a schematic diagram of the branched chain of the special structure of PPP R R R .
图7是少自由度并联机构的雅可比快速构造的流程图。 Fig. 7 is a flow chart of the fast Jacobian construction of a parallel mechanism with few degrees of freedom.
图8是3-RPS少自由度并联机构示意图。 Fig. 8 is a schematic diagram of a 3- RPS parallel mechanism with few degrees of freedom.
图中：1、球副，2、带驱动器的移动副，3、万向副，4、转动副，5、移动副， 6、带驱动器的转动副。 In the figure: 1. Ball pair, 2. Moving pair with driver, 3. Universal pair, 4. Rotating pair, 5. Moving pair, 6. Rotating pair with driver. the
具体实施方式 Detailed ways
下面通过实施例结合附图进一步说明本发明。 The present invention is further illustrated below by means of embodiments in conjunction with the accompanying drawings.
实施例1： Example 1:
如图1所示，少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个球副，且移动副的运动方向通过2个球副的中心。 As shown in Figure 1, the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two spherical pairs at its two ends, and the moving direction of the moving pair passes through the centers of the two spherical pairs.
如图2所示，少自由度并联机构的支链中含有1个万向副带驱动器的移动副和位于其两端的1个球副以及1个万向副，且移动副的运动方向通过球副的中心和万向副的中心。 As shown in Figure 2, the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a universal drive, a ball pair and a universal pair at both ends, and the moving direction of the moving pair passes through the ball The center of the vice and the center of the universal vice.
如图3所示，少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个万向副，且移动副的运动方向通过2个万向的中心。 As shown in Figure 3, the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two universal joints at its two ends, and the moving direction of the moving pair passes through the centers of the two universal joints.
如图4所示，少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的1个万向副和1个球副，且移动副的运动方向通过球副的中心和转动副的轴线。 As shown in Figure 4, the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver, a universal pair and a ball pair at both ends, and the movement direction of the moving pair passes through the center of the ball pair and the axis of the revolving pair.
如图5所示，少自由度并联机构的支链中含有1个带驱动器的移动副和位于其两端的2个移动副，且带驱动器的移动副的运动方向垂直于另外2个移动副的运动方向。 As shown in Figure 5, the branch chain of the parallel mechanism with few degrees of freedom contains a moving pair with a driver and two moving pairs located at its two ends, and the moving direction of the moving pair with a driver is perpendicular to that of the other two moving pairs. direction of motion.
如图6所示，少自由度并联机构的支链中含有1个带驱动器的转动副，带驱动器的转动副一端串联有3个移动副和2个转动副，且带驱动器的转动副的轴线垂直于另外2个转动副的轴线。 As shown in Figure 6, the branch chain of the parallel mechanism with few degrees of freedom contains a revolving pair with a driver, and one end of the revolving pair with a driver is connected in series with 3 moving pairs and 2 revolving pairs, and the axis of the revolving pair with a driver Perpendicular to the axes of the other 2 rotating pairs.
下面结合3自由度的3-RPS空间并联机构对本发明作进一步说明。含特殊支链结构的少自由度并联机构的雅可比快速构造流程图见图7所示，包括以下关键步骤： The present invention will be further described below in conjunction with the 3- RPS spatial parallel mechanism with 3 degrees of freedom. The Jacobian rapid construction flow chart of the parallel mechanism with few degrees of freedom with special branched chain structure is shown in Fig. 7, including the following key steps:
(1) 分析少自由度并联机构的支链类型； (1) Analyze the branch chain types of the parallel mechanism with few degrees of freedom;
(2) 计算支链的驱动力螺旋； (2) Calculate the driving force spiral of the branch chain;
(3) 构造少自由度并联机构的部分雅可比矩阵； (3) Construct the partial Jacobian matrix of the parallel mechanism with few degrees of freedom;
(4) 计算支链的约束力螺旋； (4) Calculate the binding force spiral of the branch chain;
(5) 构造少自由度并联机构的整体雅可比矩阵。 (5) Construct the overall Jacobian matrix of the parallel mechanism with few degrees of freedom.
以图8所示的3自由度的3-RPS空间并联机构为例。3-RPS并联机构有3条支链，每条支链含有1个转动副、1个带驱动器的移动副、1个球副。 Take the 3-DOF 3- RPS space parallel mechanism shown in Fig. 8 as an example. The 3-R P S parallel mechanism has 3 branch chains, and each branch chain contains 1 revolving pair, 1 moving pair with driver, and 1 ball pair.
(1) 分析3-RPS少自由度并联机构的支链类型。因为各支链中的驱动器布置在移动副，且驱动器的移动方向通过转动副和球副（看作3个转动副）的轴线，这种支链结构是图4所示的RPS特殊支链结构，因此可以采用本方法进行雅可比矩阵的快速构造。 (1) Analyze the branch chain types of the 3- RPS parallel mechanism with few degrees of freedom. Because the driver in each branch chain is arranged on the moving pair, and the moving direction of the driver passes through the axes of the rotating pair and the ball pair (as three rotating pairs), this kind of branch chain structure is the RPS special branch shown in Figure 4. Chain structure, so this method can be used to quickly construct the Jacobian matrix.
(2) 计算各条RPS支链的驱动力螺旋。根据图8，驱动力螺旋为 (2) Calculate the driving force helix of each RPS branch. According to Figure 8, the driving force spiral is
其中i = 1, 2, 3，为图8中沿A i B i 方向的单位矢量，是图8中坐标原点O到A i 的距离矢量，是图8中坐标原点O到B i 的距离矢量。 where i = 1, 2, 3, is the unit vector along the A i B i direction in Fig. 8, is the distance vector from the coordinate origin O to A i in Fig. 8, is the distance vector from the coordinate origin O to Bi in Figure 8.
(3) 构造3-RPS少自由度并联机构的部分雅可比矩阵。根据(式1)，3-RPS并联机构的部分雅可比矩阵为 (3) Construct the partial Jacobian matrix of the 3- RPS parallel mechanism with few degrees of freedom. According to (Equation 1), the partial Jacobian matrix of the 3- RP S parallel mechanism is
(4) 计算3-RPS少自由度并联机构的支链约束力螺旋。根据图8，支链约束力螺旋为 (4) Calculate the branch-chain constraint helix of the 3- RPS parallel mechanism with few degrees of freedom. According to Figure 8, the helix of the branched chain binding force is
其中为图8中转动副A i 轴线方向的单位矢量。 in is the unit vector in the axial direction of the rotating pair A i in Fig. 8.
(5) 构造3-RPS少自由度并联机构的整体雅可比矩阵。根据(式2)，3-RPS并联机构的整体雅可比矩阵为 (5) Construct the overall Jacobian matrix of the 3- RPS parallel mechanism with few degrees of freedom. According to (Equation 2), the overall Jacobian matrix of the 3- RPS parallel mechanism is
其中是图8中动平台中心到B i 的距离矢量。可以看出，上式与Joshi和Tsai于2002年在《ASME Journal of Mechanical Design》（124(6):第254–258页）的论文“Jacobian analysis of limited-DOF parallel manipulators”中获得的雅可比矩阵结果一致，而本方法的构造过程更快速简洁。 in is the distance vector from the center of the moving platform to Bi in Figure 8. It can be seen that the above formula is consistent with the Jacobian obtained by Joshi and Tsai in the paper "Jacobian analysis of limited-DOF parallel manipulators" in "ASME Journal of Mechanical Design" (124(6): pp. 254–258) in 2002 The matrix results are consistent, and the construction process of this method is faster and simpler.
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