This paper demonstrates a novel geometric modeling and computational method of the family of spatial parallel mechanisms (PMs) with 3-R(P)S structure for direct kinematic analysis based on the point pair relationship. The point pair relationship, which is derived from the framework of conformal geometric algebra (CGA), consists of the relationship between the point and the point pair and two point pairs. The first research is on the distance relationship between the point and the point pair. Second, the derivation of the distance relationship between two point pairs is based on the aforementioned result, which shows the mathematical homogeneity. Third, two formulations for a point of the point pairs that satisfy the distance relationship between two point pairs are reduced. Fourth, the point pair relationship is applied to solve the direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure. Finally, four numerical examples are provided to verify the validity of the proposed algorithm. Overall, the proposed method can be generalized for the direct kinematics of a series of spatial parallel mechanisms with 3-R(P)S structure.

This paper proposes a variable stiffness joint based on a symmetrical crank slider mechanism (SCM-VSJ). Firstly, the mechanical design and the working principle of the variable stiffness joint is described, and its stiffness regulation characteristics are studied. Secondly, the dynamical model of variable stiffness joint including joint motor, harmonic reducer and stiffness adjustment motor is established, in addition, the transmission mechanism of the crank slider mechanism and the elastic deformation of the spring bar are considered in the dynamic modeling. Finally, in order to control the dynamic stiffness of the variable stiffness joint in real time, a kind of improved PID (proportional-integral-derivative) control algorithm based on feed-forward and feedback closed-loop is proposed on the basis of the existing dynamical model, and the simulation analysis of real-time tracking control of dynamic stiffness for sinusoidal wave expected stiffness signal and random expected stiffness signal is carried out respectively. The research shows that the real-time stiffness control of SCM-VSJ can be realized effectively, and during the stiffness adjustment process, the output torque of the stiffness adjustment motor will be affected by the elastic deformation of the spring bar.

A novel geometric modeling and calculation method for forward displacement analysis of the 6-3 Stewart platforms is proposed by using the conformal geometric algebra (CGA) framework. Firstly, two formulas between 2-blade and 1-blade are formulated. Secondly, the expressions for two spherical joints of the moving platform are given via CGA operation. Thirdly, a coordinate-invariant geometric constraint equation is deduced. Fourthly, a 16-degree univariate polynomial equation without algebraic elimination by using the Euler angle substitution is presented. Fifthly, the coordinates of three spherical joints on the moving platform are calculated without judging the radical symbols. Finally, two numerical examples are used to verify the method. The highlight of this paper is that a new geometric modeling and calculation method without algebraic elimination is obtained by using the determinant form of the CGA inner product algorithm, which provides a new idea to solve a more complex spatial parallel mechanism in the future.

The nonlinear characteristics in the bicycle robot dynamic system make the bicycle robot control much more difficult. Aiming at this problem, the cascade sliding-mode control method was applied to the bicycle robot nonlinear system. A kind of bicycle robot nonlinear dynamic model presented based on Lagrange method was analyzed and the cascade sliding-mode controller of multi-sliding surface was designed with the method of variable structure control theory. This method divided system states into two sub-systems. Firstly, two state variables of subsystem are chosen to construct the first-layer sliding surface, on the basis of the first-layer sliding function, the first layer sliding surface and one of the residual state variables are used to construct the second-layer sliding surface, at last, all of the system states are included in final sliding surface. The ultimate control is obtained by using Lyapunov stability theorem. since each sub-sliding surface contains the equivalent components of sliding surface, which makes the system states move along the each sub-sliding surface. The cascade sliding-mode controller has the bicycle robot controlled effectively, which has been verified in the simulation experiment.