Optimization control method for shock absorber

[0002] The present application claims priority benefit of Japanese Application No. Hei 11-009954, filed Jan. 18, 1999, the disclosure of which is hereby included by reference in its entirety.

[0003] 1. Field of the Invention

[0004] This invention relates to an optimization control method for a shock absorber having a non-linear kinetic characteristic.

[0005] 2. Description of the Related Art

[0006] Feedback control systems are widely used to maintain the output of a dynamic system at a desired value in spite of external disturbance forces that would move the output away from the desired value. For example, a household furnace controlled by a thermostat is an example of a feedback control system. The thermostat continuously measures the air temperature of the house, and when the temperature falls below a desired minimum temperature, the thermostat turns the furnace on. When the furnace has warmed the air above the desired minimum temperature, then the thermostat turns the furnace off. The thermostat-furnace system maintains the household temperature at a constant value in spite of external disturbances such as a drop in the outside air temperature. Similar types of feedback control are used in many applications.

[0007] A central component in a feedback control system is a controlled object, otherwise known as a process “plant,” whose output variable is to be controlled. In the above example, the plant is the house, the output variable is the air temperature of the house, and the disturbance is the flow of heat through the walls of the house. The plant is controlled by a control system. In the above example, the control system is the thermostat in combination with the furnace. The thermostat-furnace system uses simple on-off feedback control to maintain the temperature of the house. In many control environments, such as motor shaft position or motor speed control systems, simple on-off feedback control is insufficient. More advanced control systems rely on combinations of proportional feedback control, integral feedback control, and derivative feedback control. Feedback that is the sum of proportional plus integral plus derivative feedback is often referred to as PID control.

[0008] The PID control system is a linear control system that is based on a dynamic model of the plant. In classical control systems, a linear dynamic model is obtained in the form of dynamic equations, usually ordinary differential equations. The plant is assumed to be relatively linear, time invariant, and stable. However, many real-world plants are time varying, highly nonlinear, and unstable. For example, the dynamic model may contain parameters (e.g., masses, inductances, aerodynamic coefficients, etc.) which are either poorly known or depend on a changing environment. Under these conditions, a linear PID controller is insufficient.

[0009] Evaluating the motion characteristics of a nonlinear plant is often difficult, in part due to the lack of a general analysis method. Conventionally, when controlling a plant with nonlinear motion characteristics, it is common to find certain equilibrium points of the plant and the motion characteristics of the plant are linearized in a vicinity near an equilibrium point. Control is then based on evaluating the pseudo (linearized) motion characteristics near the equilibrium point. This technique works poorly, if at all, for plants described by models that are unstable or dissipative. The optimization control for a non-linear kinetic characteristic of a controlled process has not been well developed. A general analysis method for non-linear kinetic characteristic has not been previously available, so a control device suited for the linear-kinetic characteristic is often substituted. Namely, for the controlled process with the non-linear kinetic characteristic, a suitable balance point for the kinetic characteristic is picked. Then, the kinetic characteristic of the controlled process is linearized in a vicinity of the balance point, whereby the evaluation is conducted relative to pseudo-kinetic characteristics.

[0010] However, this method has several disadvantageous. Although the optimization control may be accurately conducted around the balance point, its accuracy decreases beyond this balance point. Further, this method cannot typically keep up with various kinds of environmental changes around the controlled process.

[0011] Shock absorbers used for automobiles and motor cycles are one example of a controlled process having the non-linear kinetic characteristic. The optimization of the non-linear kinetic characteristic has been long sought because vehicle's turning performances and ride are greatly affected by the damping characteristic and output of the shock absorbers.

[0012] The present invention solves these and other problems by providing a new control system for optimizing a shock absorber having a non-linear kinetic characteristic. The new AI control system uses a fitness (performance) function that is based on the physical laws of minimum entropy. This control system can be used to control complex plants described by nonlinear, unstable, dissipative models. The control system is configured to use smart simulation techniques for controlling the shock absorber (plant).

[0013] In one embodiment, the control system comprises a learning system, such as a neural network that is trained by a genetic analyzer. The genetic analyzer uses a fitness function that maximizes sensor information while minimizing entropy production.

[0014] In one embodiment, a suspension control uses a difference between the time differential (derivative) of entropy from the learning control unit and the time differential of the entropy inside the controlled process (or a model of the controlled process) as a measure of control performance. In one embodiment, the entropy calculation is based on a thermodynamic model of an equation of motion for a controlled process plant that is treated as an open dynamic system.

[0015] The control system is trained by a genetic analyzer. The optimized control system provides an optimum control signal based on data obtained from one or more sensors. For example, in a suspension system, a plurality of angle and position sensors can be used. In an off-line (laboratory) learning mode, fuzzy rules are evolved using a kinetic model (or simulation) of a vehicle and is suspension system. Data from the kinetic model is provided to an entropy calculator which calculates input and output entropy production of the model. The input and output entropy productions are provided to a fitness function calculator that calculates a fitness function as a difference in entropy production rates for the genetic analyzer. The genetic analyzer uses the fitness function to develop a training signal for the off-line control system. Control parameters from the off-line control system are then provided to an online control system in the vehicle.

[0016] In one embodiment, the invention includes a method for controlling a nonlinear object (a plant) by obtaining an entropy production difference between a time differentiation (dS_u/dt) of the entropy of the plant and a time differentiation (dS_c/dt) of the entropy provided to the plant from a controller. A genetic algorithm that uses the entropy production difference as a fitness (performance) function evolves a control rule in an off-line controller. The nonlinear stability characteristics of the plant are evaluated using a Lyapunov function. The genetic analyzer minimizes entropy and maximizes sensor information content. Control rules from the off-line controller are provided to an online controller to control suspension system. In one embodiment, the online controller controls the damping factor of one or more shock absorbers (dampers) in the vehicle suspension system.

[0017] In some embodiments, the control method also includes evolving a control rule relative to a variable of the controller by means of a genetic algorithm. The genetic algorithm uses a fitness function based on a difference between a time differentiation of the entropy of the plant (dS_u/dt) and a time differentiation (dS_c/dt) of the entropy provided to the plant. The variable can be corrected by using the evolved control rule.

[0018] In another embodiment, the invention comprises an AI control system adapted to control a nonlinear plant. The AI control system includes a simulator configured to use a thermodynamic model of a nonlinear equation of motion for the plant. The thermodynamic model is based on a Lyapunov function (V), and the simulator uses the function V to analyze control for a state stability of the plant. The AI control system calculates an entropy production difference between a time differentiation of the entropy of said plant (dS_u/dt) and a time differentiation (dS_c/dt) of the entropy provided to the plant by a low-level controller that controls the plant. The entropy production difference is used by a genetic algorithm to obtain an adaptation function in which the entropy production difference is minimized. The genetic algorithm provides a teaching signal to a fuzzy logic classifier that determines a fuzzy rule by using a learning process. The fuzzy logic controller is also configured to form a control rule that sets a control variable of the controller in the vehicle.

[0019] In yet another embodiment, the invention comprises a new physical measure of control quality based on minimum production entropy and using this measure for a fitness function of genetic algorithm in optimal control system design. This method provides a local entropy feedback loop in the control system. The entropy feedback loop provides for optimal control structure design by relating stability of the plant (using a Lyapunov function) and controllability of the plant (based on production entropy of the control system). The control system is applicable to all control systems, including, for example, control systems for mechanical systems, bio-mechanical systems, robotics, electro-mechanical systems, etc.

[0020] The advantages and features of the disclosed invention will readily be appreciated by persons skilled in the art from the following detailed description when read in conjunction with the drawings listed below.

[0021] FIG. 1 is a block diagram illustrating a control system for a shock absorber.

[0022] FIG. 2A is a block diagram showing a fuzzy control unit that estimates an optimal throttle amount for each shock absorber and outputs signals according to the predetermined fuzzy rule based on the detection results.

[0023] FIG. 2B is a block diagram showing a learning control unit having a fuzzy neural network.

[0024] FIG. 3 is a schematic diagram of a four-wheel vehicle suspension system showing the parameters of the kinetic models for the vehicle and suspension system.

[0025] FIG. 4 is a detailed view of the parameters associated with the right-front wheel from FIG. 3.

[0026] FIG. 5 is a detailed view of the parameters associated with the left-front wheel from FIG. 3.

[0027] FIG. 6 is a detailed view of the parameters associated with the right-rear wheel from FIG. 3.

[0028] FIG. 7 is a detailed view of the parameters associated with the left-rear wheel from

[0029] FIG. 1 is a block diagram illustrating one embodiment of an optimization control system 100 for controlling one or more shock absorbers in a vehicle suspension system.

[0030] This control system 100 is divided in an actual (online) control module 102 in the vehicle and a learning (offline) module 101. The learning module 101 includes a learning controller 118, such as, for example, a fuzzy neural network (FNN). The learning controller (hereinafter “the FNN 118”) can be any type of control system configured to receive a training input and adapt a control strategy using the training input. A control output from the FNN 118 is provided to a control input of a kinetic model 120 and to an input of a first entropy production calculator 116. A sensor output from the kinetic model is provided to a sensor input of the FNN 118 and to an input of a second entropy production calculator 114. An output from the first entropy production calculator 116 is provided to a negative input of an adder 119 and an output from the second entropy calculator 114 is provided to a positive input of the adder 119. An output from the adder 119 is provided to an input of a fitness (performance) function calculator 112. An output from the fitness function calculator 112 is provided to an input of a genetic analyzer 110. A training output from the genetic analyzer 110 is provided to a training input of the FNN 118.

[0031] The actual control module 102 includes a fuzzy controller 124. A control-rule output from the FNN 118 is provided to a control-rule input of a fuzzy controller 124. A sensor-data input of the fuzzy controller 124 receives sensor data from a suspension system 126. A control output from the fuzzy controller 124 is provided to a control input of the suspension system 126. A disturbance, such as a road-surface signal, is provided to a disturbance input of the kinetic model 120 and to the vehicle and suspension system 126.

[0032] The actual control module 102 is installed into a vehicle and controls the vehicle suspension system 126. The learning module 101 optimizes the actual control module 102 by using the kinetic model 120 of the vehicle and the suspension system 126. After the learning control module 101 is optimized by using a computer simulation, one or more parameters from the FNN 118 are provided to the actual control module 101.

[0033] In one embodiment, a damping coefficient control-type shock absorber is employed, wherein the fuzzy controller 124 outputs signals for controlling a throttle in an oil passage in one or more shock absorbers in the suspension system 126.

[0034] FIGS. 2A and 2B illustrate one embodiment of a fuzzy controller 200 suitable for use in the FNN 118 and/or the fuzzy controller 124. In the fuzzy controller 200, data from one or more sensors is provided to a fuzzification interface 204. An output from the fuzzification interface 204 is provided to an input of a fuzzy logic module 206. The fuzzy logic module 206 obtains control rules from a knowledge-base 202. An output from the fuzzy logic module 206 is provided to a de-fuzzification interface 208. A control output from the de-fuzzification interface 208 is provided to a controlled process 210 (e.g. the suspension system 126, the kinetic model 120, etc.).

[0035] The sensor data shown in FIGS. 1 and 2, can include, for example, vertical positions of the vehicle z₀, pitch angle β, roll angle α, suspension angle η for each wheel, arm angle θ for each wheel, suspension length z₆for each wheel, and/or deflection z₁₂for each wheel. The fuzzy control unit estimates the optimal throttle amount for each shock absorber and outputs signals according to the predetermined fuzzy rule based on the detection results.

[0036] The learning module 101 includes a kinetic model 120 of the vehicle and suspension to be used with the actual control module 101, a learning control module 118 having a fuzzy neural network corresponding to the actual control module 101 (as shown in FIG. 2B), and an optimizer module 115 for optimizing the learning control module 118.

[0037] The optimizer module 115 computes a difference between a time differential of entropy from the FNN 118 (dSc/dt) and a time differential of entropy inside the subject process (i.e., vehicle and suspensions) obtained from the kinetic model 120. The computed difference is used as a performance function by a genetic optimizer 110. The genetic optimizer 110 optimizes (trains) the FNN 118 by genetically evolving a teaching signal. The teaching signal is provided to a fuzzy neural network in the FNN 118. The genetic optimizer 110 optimizes the fuzzy neural network (FNN) such that an output of the FNN, when used as an input to the kinetic module 120, reduces the entropy difference between the time differentials of both entropy values.

[0038] The fuzzy rules from the FNN 118 are then provided to a fuzzy controller 124 in the actual control module 102. Thus, the fuzzy rule (or rules) used in the fuzzy controller 124 (in the actual control module 101), are determined based on an output from the FNN 118 (in the learning control unit), that is optimized by using the kinetic model 120 for the vehicle and suspension.

[0039] The genetic algorithm 110 evolves an output signal a based on a performance function 𝑓. Plural candidates for α are produced and these candidates are paired according to which plural chromosomes (parents) are produced. The chromosomes are evaluated and sorted from best to worst by using the performance function 𝑓. After the evaluation for all parent chromosomes, good offspring chromosomes are selected from among the plural parent chromosomes, and some offspring chromosomes are randomly selected. The selected chromosomes are crossed so as to produce the parent chromosomes for the next generation. Mutation may also be provided. The second-generation parent chromosomes are also evaluated (sorted) and go through the same evolutionary process to produce the next-generation (i.e., third-generation) chromosomes. This evolutionary process is continued until it reaches a predetermined generation or the evaluation function 𝑓 finds a chromosome with a certain value. The outputs of the genetic algorithm are the chromosomes of the last generation. These chromosomes become input information α provided to the FNN 118.

[0040] In the FNN 118, a fuzzy rule to be used in the fuzzy controller 124 is selected from a set of rules. The selected rule is determined based on the input information α from the genetic algorithm 110. Using the selected rule, the fuzzy controller 124 generates a control signal C_dnfor the vehicle and suspension system 126. The control signal adjusts the operation (damping factor) of one or more shock absorbers to produce a desired ride and handling quality for the vehicle.

[0041] The genetic algorithm 110 is a nonlinear optimizer that optimizes the performance function 𝑓. As is the case with most optimizers, the success or failure of the optimization often ultimately depends on the selection of the performance function 𝑓.

[0042] The fitness function 112 𝑓 for the genetic algorithm 110 is given by $\begin{array}{cc}f=\min \frac{dS}{dt}& \left(1\right)\end{array}$

[0043] where $\begin{array}{cc}\frac{dS}{dt}=(\frac{d{S}_c}{dt}-\frac{d{S}_u}{dt})& \left(2\right)\end{array}$

[0044] The quantity dS_u/dt represents the rate of entropy production in the output x(t) of the kinetic model 120. The quantity dS_c/dt represents the rate of entropy production in the output C_dnof the FNN 118.

[0045] Entropy is a concept that originated in physics to characterize the heat, or disorder, of a system. It can also be used to provide a measure of the uncertainty of a collection of events, or, for a random variable, a distribution of probabilities. The entropy function provides a measure of the lack of information in the probability distribution. To illustrate, assume that p(x) represents a probabilistic description of the known state of a parameter, that p(x) is the probability that the parameter is equal to z. If p(x) is uniform, then the parameter p is equally likely to hold any value, and an observer will know little about the parameter p. In this case, the entropy function is at its maximum. However, if one of the elements of p(z) occurs with a probability of one, then an observer will know the parameter p exactly and have complete information about p. In this case, the entropy of p(x) is at its minimum possible value. Thus, by providing a measure of uniformity, the entropy function allows quantification of the information on a probability distribution.

[0046] It is possible to apply these entropy concepts to parameter recovery by maximizing the entropy measure of a distribution of probabilities while constraining the probabilities so that they satisfy a statistical model given measured moments or data. Though this optimization, the distribution that has the least possible information that is consistent with the data may be found. In a sense, one is translating all of the information in the data into the form of a probability distribution. Thus, the resultant probability distribution contains only the information in the data without imposing additional structure. In general, entropy techniques are used to formulate the parameters to be recovered in terms of probability distributions and to describe the data as constraints for the optimization. Using entropy formulations, it is possible to perform a wide range of estimations, address ill-posed problems, and combine information from varied sources without having to impose strong distributional assumptions.

[0047] Entropy-based optimization of the FNN is based on obtaining the difference between a time differentiation (dS_u/dt) of the entropy of the plant and a time differentiation (dS_c/dt) of the entropy provided to the kinetic model from the FNN 118 controller that controls the kinetic model 120, and then evolving a control rule using a genetic algorithm. The time derivative of the entropy is called the entropy production rate. The genetic algorithm 110 minimizes the difference between the entropy production rate of the kinetic model 120 (that is, the entropy production of the controlled process) (dS_u/dt) and the entropy production rate of the low-level controller (dS_c/dt) as a performance function. Nonlinear operation characteristics of the kinetic model (the kinetic model represents a physical plant) are calculated by using a Lyapunov function.

[0048] The dynamic stability properties of the model 120 near an equilibrium point can be determined by use of Lyapunov functions as follows. Let V(x) be a continuously differentiable scalar function defined in a domain D⊂Rⁿthat contains the origin. The function V(x) is said to be positive definite if V(0)=0 and V(x)>0 for x≠0. The function V(x) is said to be positive semidefinite if V(x)≧0 for all x. A function V(x) is said to be negative definite or negative semidefinite if −V(x) is positive definite or positive semidefinite, respectively. The derivative of V along the trajectories {dot over (x)}=𝑓(x) is given by: $\begin{array}{cc}\stackrel{.}{V}\left(x\right)=\sum _{i=1}^n\frac{\partial V}{\partial x_i}{\stackrel{.}{x}}_i=\frac{\partial V}{\partial x}f\left(x\right)& \left(3\right)\end{array}$

[0049] where ∂V/∂x is a row vector whose ith component is ∂V/∂x_iand the components of the n-dimensional vector 𝑓(x) are locally Lipschitz functions of x, defined for all x in the domain D. The Lyapunov stability theorem states that the origin is stable if there is a continuously differentiable positive definite function V(x) so that {dot over (V)}(x) is negative definite. A function V(x) satisfying the conditions for stability is called a Lyapunov function.

[0050] The genetic algorithm realizes 110 the search of optimal controllers with a simple structure using the principle of minimum entropy production.

[0051] The fuzzy tuning rules are shaped by the learning system in the fuzzy neural network 118 with acceleration of fuzzy rules on the basis of global inputs provided by the genetic algorithm 110.

[0052] In general, the equation of motion for non-linear systems is expressed as follows by defining “q” as generalized coordinates, “f” and “g” random functions, “Fe” as control input. $\begin{array}{cc}q=f\left(\stackrel{.}{q},q\right)+g\left(q\right)-F_e& \left(a\right)\end{array}$

[0053] In the above equation, when the dissipation term and control input in the second term are multiplied by a speed, the following equation can be obtained for the time differentials of the entropy. $\begin{array}{cc}\frac{dS}{dt}=f\left(\stackrel{.}{q},q\right)\stackrel{.}{q}-\mathrm{Feq}=\frac{d{S}_u}{dt}-\frac{d{S}_c}{dt}& \left(b\right)\end{array}$

[0054] dS/dt is a time differential of entropy for the entire system. dS_u/dt is a time differential of entropy for the plant, that is the controlled process. dS_c/dt is a time differential of entropy for the control system for the plant.

[0055] The following equation is selected as Lyapunov function for the equation (a). $\begin{array}{cc}V=\left(\sum q^2+S^2\right)/2=\left(\sum q^2+{\left(S_u-S_c\right)}^2\right)/2& \left(c\right)\end{array}$

[0056] The greater the integral of the Lyapunov function, the more stable the kinetic characteristic of the plant.

[0057] Thus, for the stabilization of the systems, the following equation is introduced as a relationship between the Lyapunov function and entropy production for the open dynamic system. $\begin{array}{cc}DV/dt=\sum qq+\left(S_u-S_c\right)\left(d{S}_u/dt-d{S}_c/dt\right)<0& \left(d\right)\\ \sum qq<\left(S_u-S_c\right)\left(d{S}_c/dt-d{S}_u/dt\right)& \left(e\right)\end{array}$

[0058] A Duffing oscillator is one example of a dynamic system. In the Duffing oscillator, the equation of motion is expressed as: $\begin{array}{cc}\ddot{x}+\stackrel{.}{x}+x+x^3=0& \left(f\right)\end{array}$

[0059] the entropy production from this equation is calculated as: $\begin{array}{cc}dS/dt=x^3& \left(g\right)\end{array}$

[0060] Further, Lyapunov function relative to the equation (f) becomes: $\begin{array}{cc}V=\left(1/2\right)x^2+U\left(x\right),U\left(x\right)=\left(1/4\right)x^4-\left(1/2\right)x^2& \left(h\right)\end{array}$

[0061] If the equation (f) is modified by using the equation (h), it is expressed as: $\begin{array}{cc}\ddot{x}+x+\frac{\partial U\left(x\right)}{\partial x}=0& \left(i\right)\end{array}$

[0062] If the left side of the equation (i) is multiplied by x as: $\ddot{x}+x+\frac{\partial U\left(x\right)}{\partial x}x=0$

[0063] Then, if the Lyapunov function is differentiated by time, it becomes: $dV/dt=xx+\left(\partial U\left(x\right)/\partial x\right)x$

[0064] If this is converted to a simple algebra, it becomes: $\begin{array}{cc}dV/dt=\left(1/T\right)\left(dS/dt\right)& \left(j\right)\end{array}$

[0065] wherein “T” is a normalized factor.

[0066] dS/dt is used for evaluating the stability of the system. dS_u/dt is a time change of the entropy for the plant. −dS_c/dt is considered to be a time change of negative entropy given to the plant from the control system.

[0067] The present invention calculates waste such as disturbances for the entire control system of the plant based on a difference between the time differential dS_u/dt of the entropy of the plant that is a controlled process and time differential dSu/dt of the entropy of the plant. Then, the evaluation is conducted by relating to the stability of the controlled process that is expressed by Lyapunov function. In other words, the smaller the difference of both entropy, the more stable the operation of the plants.

[0068] Suspension Control

[0069] In one embodiment, the control system 100 of FIGS. 1-2 is applied to a suspension control system, such as, for example, in an automobile, truck, tank, motorcycle, etc.

[0070] FIG. 3 is a schematic diagram of an automobile suspension system. In FIG. 3, a right front wheel 301 is connected to a right arm 313. A spring and damper linkage 334 controls the angle of the arm 313 with respect to a body 310. A left front wheel 302 is connected to a left arm 323 and a spring and damper 324 controls the angle of the arm 323. A front stabilizer 330 controls the angle of the left arm 313 with respect to the right arm 323. Detail views of the four wheels are shown in FIG. 4-7. Similar linkages are shown for a right rear wheel 303 and a left rear wheel 304. The

[0071] In one embodiment of the suspension control system, the learning module 101 uses a kinetic model 120 for the vehicle and suspension. FIG. 3 illustrates each parameter of the kinetic models for the vehicle and suspensions. FIGS. 4-7 illustrate exploded views for each wheel as illustrated in FIG. 3.

[0072] A kinetic model 120 for the suspension system in the vehicle 300 shown in FIGS. 3-7 is developed as follows.

[0073] 1. Description of Transformation Matrices

[0074] 1.1 A Global Reference Coordinate x_r, y_r, z_r{r} is Assumed to be at the Geometric Center P_rof the Vehicle Body 310

[0075] The following are the transformation matrices to describe the local coordinates for:

[0076] {2} is a local coordinate in which an origin is the center of gravity of the vehicle body 310;

[0077] {7} is a local coordinate in which an origin is the center of gravity of the suspension;

[0078] {10n} is a local coordinate in which an origin is the center of gravity of the n'th arm;

[0079] {12n} is a local coordinate in which an origin is the center of gravity of the n'th wheel;

[0080] {13n} is a local coordinate in which an origin is a contact point of the n'th wheel relative to the road surface; and

[0081] {14} is a local coordinate in which an origin is a connection point of the stabilizer.

[0082] Note that in the development that follows, the wheels 302, 301, 304, and 303 are indexed using “i”, “ii”, “iii”, and “iv”, respectively.

[0083] 1.2 Transformation Matrices

[0084] As indicated, “n” is a coefficient indicating wheel positions such as i, ii, iii, and iv for left front, right front, left rear and right rear respectively. The local coordinate systems x₀, y₀, and z₀{0} are expressed by using the following conversion matrix that moves the coordinate {r} along a vector (0,0,z₀) ${}_0^{r}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_0\\ 0& 0& 0& 1\end{array}\right]$

[0085] Rotating the vector {r} along y_rwith an angle β makes a local coordinate system x_0c, y_0c, z_0c{0r} with a transformation matrix _0c⁰T. $\begin{array}{cc}{}_{0c}^{0}T=\left[\begin{array}{cccc}\cos \beta & 0& \sin \beta & 0\\ 0& 1& 0& 0\\ -\sin \beta & 0& \cos \beta & 0\\ 0& 0& 0& 1\end{array}\right]& \left(4\right)\end{array}$

[0086] Transferring {0r} through the vector (a_1n, 0, 0) makes a local coordinate system x_0f, y_0f, z_0f{0f} with a transformation matrix ^0r_0fT. $\begin{array}{cc}{}_{0n}^{0c}T=\left[\begin{array}{cccc}1& 0& 0& a_{1n}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(5\right)\end{array}$

[0087] The above procedure is repeated to create other local coordinate systems with the following transformation matrices. $\begin{array}{cc}{}_{1n}^{0n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]& \left(6\right)\end{array}$ $\begin{array}{cc}{}_2^{1i}T=\left[\begin{array}{cccc}1& 0& 0& a_0\\ 0& 1& 0& b_0\\ 0& 0& 1& c_0\\ 0& 0& 0& 1\end{array}\right]& \left(7\right)\end{array}$

[0088] 1.3 Coordinates for the Wheels (Index n: i for the Left Front, ii for the Right Front, etc.) are Generated as Follows

[0089] Transferring {1n} through the vector (0, b_2n, 0) makes local coordinate system x_3n, y_3n, z_3n{3n} with transformation matrix ^1f_3nT. $\begin{array}{cc}{}_{3n}^{1n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& b_{2n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(8\right)\end{array}$ $\begin{array}{cc}{}_{4n}^{3n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\gamma }_n& -\sin {\gamma }_n& 0\\ 0& \sin {\gamma }_n& \cos {\gamma }_n& 0\\ 0& 0& 0& 1\end{array}\right]& \left(9\right)\end{array}$ $\begin{array}{cc}{}_{5n}^{4n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& c_{1n}\\ 0& 0& 0& 1\end{array}\right]& \left(10\right)\end{array}$ $\begin{array}{cc}{}_{6n}^{5n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\eta }_n& -\sin {\eta }_n& 0\\ 0& \sin {\eta }_n& \cos {\eta }_n& 0\\ 0& 0& 0& 1\end{array}\right]& \left(11\right)\end{array}$ $\begin{array}{cc}{}_{7n}^{6n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_{6n}\\ 0& 0& 0& 1\end{array}\right]& \left(12\right)\end{array}$ $\begin{array}{cc}{}_{8n}^{4n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& c_{2n}\\ 0& 0& 0& 1\end{array}\right]& \left(13\right)\end{array}$ $\begin{array}{cc}{}_{9n}^{8n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& 0\\ 0& \sin {\theta }_n& \cos {\theta }_n& 0\\ 0& 0& 0& 1\end{array}\right]& \left(14\right)\end{array}$ $\begin{array}{cc}{}_{10n}^{9n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{1n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(15\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{11n}^{9n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{3n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(16\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{12n}^{11n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\cos \zeta }_n& -{\sin \zeta }_n& 0\\ 0& {\sin \zeta }_n& {\cos \zeta }_n& 0\\ 0& 0& 0& 1\end{array}\right]& \left(17\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{13n}^{12n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_{12n}\\ 0& 0& 0& 1\end{array}\right]& \left(18\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{14n}^{9n}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{0n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]& \left(19\right)\end{array}$

[0090] 1.4 Some Matrices are Sub-Assembled to Make the Calculation Simpler $\begin{array}{cc}{\hspace{0.17em}}_{1n}^{r}T={\hspace{0.17em}}_0^{r}T{\hspace{0.17em}}_{0n}^{0c}T{\hspace{0.17em}}_{1n}^{0n}T\begin{array}{c}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \beta & 0& \sin \beta & 0\\ 0& 1& 0& 0\\ -\sin \beta & 0& \cos \beta & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& a_{1n}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}\cos \beta & 0& \sin \beta & a_{1n}\cos \beta \\ 0& 1& 0& 0\\ -\sin \beta & 0& \cos \beta & z_0-a_1\sin \beta \\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}\cos \beta & \sin \mathrm{\beta sin\alpha }& \sin \mathrm{\beta cos\alpha }& a_{1n}\cos \beta \\ 0& \cos \alpha & -\sin \alpha & 0\\ -\sin \beta & \cos \mathrm{\beta sin\alpha }& \cos \mathrm{\beta cos\alpha }& z_0-a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\end{array}& \left(20\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{4n}^{r}T={\hspace{0.17em}}_{1n}^{r}T{\hspace{0.17em}}_{3n}^{1n}T{\hspace{0.17em}}_{4n}^{3n}T\begin{array}{c}=\left[\begin{array}{cccc}\cos \beta & \sin \mathrm{\beta sin\alpha }& \sin \mathrm{\beta cos\alpha }& a_{1n}\cos \beta \\ 0& \cos \alpha & -\sin \alpha & 0\\ -\sin \beta & \cos \mathrm{\beta sin\alpha }& \cos \mathrm{\beta cos\alpha }& z_0-a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& b_{2n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]·\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{cos\gamma }}_n& -{\mathrm{sin\gamma }}_n& 0\\ 0& {\mathrm{sin\gamma }}_n& {\mathrm{cos\gamma }}_n& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}\cos \beta & \sin \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \sin \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& b_{2n}\sin \mathrm{\beta sin\alpha }+a_{1n}\cos \beta \\ 0& \cos \left(\alpha +{\gamma }_n\right)& -\sin \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \alpha \\ -\sin \beta & \cos \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \cos \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& z_0-b_{2n}c\mathrm{os}\mathrm{\beta sin\alpha }-a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\end{array}& \left(21\right)\end{array}$ $\begin{array}{cc}{\hspace{0.17em}}_{7n}^{4n}T={\hspace{0.17em}}_{5n}^{4n}T{\hspace{0.17em}}_{6n}^{5n}T{\hspace{0.17em}}_{7n}^{6n}T\begin{array}{c}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& c_{1n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{cos\eta }}_n& -{\mathrm{sin\eta }}_n& 0\\ 0& {\mathrm{sin\eta }}_n& {\mathrm{cos\eta }}_n& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_{6n}\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{cos\eta }}_n& -{\mathrm{sin\eta }}_n& 0\\ 0& {\mathrm{sin\eta }}_n& {\mathrm{cos\eta }}_n& c_{1n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_{6n}\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{cos\eta }}_n& -{\mathrm{sin\eta }}_n& -z_{6n}{\mathrm{sin\eta }}_n\\ 0& {\mathrm{sin\eta }}_n& {\mathrm{cos\eta }}_n& c_{1n}+z_{6n}{\mathrm{cos\eta }}_n\\ 0& 0& 0& 1\end{array}\right]\end{array}& \left(22\right)\end{array}$ $\begin{array}{cc}\begin{array}{c}{}_{10n}^{4n}T={}_{8n}{}^{4n}T_{9n}^{8n}{T}_{10n}^{9n}T\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& c_{2n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& 0\\ 0& \sin {\theta }_n& \cos {\theta }_n& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{1n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& 0\\ 0& \sin {\theta }_n& \cos {\theta }_n& c_{2n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{1n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& e_{1n}\cos {\theta }_n\\ 0& \sin {\theta }_n& \cos {\theta }_n& c_{2n}+e_{1n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\end{array}& \left(23\right)\end{array}$ $\begin{array}{cc}\begin{array}{c}{}_{12n}^{4n}T={}_{8n}{}^{4n}T_{9n}^{8n}{T}_{11n}^{9n}{T}_{12n}^{11n}T\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& c_{2n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& 0\\ 0& \sin {\theta }_n& \cos {\theta }_n& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{3n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\zeta }_n& -\sin {\zeta }_n& 0\\ 0& \sin {\zeta }_n& \cos {\zeta }_n& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& 0\\ 0& \sin {\theta }_n& \cos {\theta }_n& c_{2n}\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& e_{3n}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\zeta }_n& -\sin {\zeta }_n& 0\\ 0& \sin {\zeta }_n& \cos {\zeta }_n& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& e_{3n}\cos {\theta }_n\\ 0& \sin {\theta }_n& \cos {\theta }_n& c_{2n}+e_{3n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\zeta }_n& -\sin {\zeta }_n& 0\\ 0& \sin {\zeta }_n& \cos {\zeta }_n& 0\\ 0& 0& 0& 1\end{array}\right]\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \left({\theta }_n+{\zeta }_n\right)& -\sin \left({\theta }_n+{\zeta }_n\right)& e_{3n}\cos {\theta }_n\\ 0& \sin \left({\theta }_n+{\zeta }_n\right)& \cos \left({\theta }_n+{\zeta }_n\right)& c_{2n}+e_{3n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\end{array}& \left(24\right)\end{array}$

[0091] 2. Description of All the Parts of the Model Both in Local Coordinate Systems and Relations to the Coordinate {r} or {1n} Referenced to the Vehicle Body 310

[0092] 2.1 Description in Local Coordinate Systems $\begin{array}{cc}{P}_{\mathrm{body}}^2=P_{\mathrm{susp}.n}^{7n}=P_{\mathrm{arm}.n}^{10n}=P_{\mathrm{wheel}.n}^{12n}=P_{\mathrm{touchpoint}.n}^{13n}=P_{\mathrm{stab}.n}^{14n}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]& \left(25\right)\end{array}$

[0093] 2.2 Description in Global Reference Coordinate System {r} $\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{body}}^r={}_{1i}{}^{r}T_2^{1i}{\mathrm{TP}}_{\mathrm{body}}^2\\ =\left[\begin{array}{cccc}\cos \beta & \sin \beta \sin \alpha & \sin \beta \cos \alpha & a_{1i}\cos \beta \\ 0& \cos \alpha & -\sin \alpha & 0\\ -\sin \beta & \cos \beta \sin \alpha & \cos \beta \cos \alpha & z_0-a_{1i}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& a_0\\ 0& 1& 0& b_0\\ 0& 0& 1& c_0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ =[\begin{array}{c}{a}_0\cos \beta +b_0\sin \beta \sin \alpha +c_0\sin \beta \cos \alpha +a_{1i}\cos \beta \\ b_0\cos \alpha -c_0\sin \alpha \\ -a_0\sin \beta +b_0\cos \beta \sin \alpha +c_0\cos \beta \cos \alpha -a_{1i}\sin \beta \\ 1\end{array}]\end{array}& \left(26\right)\end{array}$ $\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{susp}.n}^r={}_{4n}{}^{r}T_{7n}^{4n}{\mathrm{TP}}_{\mathrm{suspn}}^{7n}\\ =\left[\begin{array}{cccc}\cos \beta & \sin \beta \sin \left(\alpha +{\gamma }_n\right)& \sin \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& b_{2n}\sin \beta \sin \alpha +a_{1n}\cos \beta \\ 0& \cos \left(\alpha +{\gamma }_n\right)& -\sin \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \alpha \\ -\sin \beta & \cos \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \cos \beta \cos \left(\alpha +{\gamma }_n\right)& z_0+b_{2n}\cos \mathrm{\beta sin}\alpha -a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]·\\ \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\eta }_n& -\sin {\eta }_n& -z_{6n}\sin {\eta }_n\\ 0& \sin {\eta }_n& \cos {\eta }_n& c_{1n}+z_{6n}\cos {\eta }_n\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ =[\begin{array}{c}\left\{z_{6n}\cos \left(\alpha +{\gamma }_n+{\eta }_n\right)+c_{1n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\sin \beta +a_{1n}\cos \beta \\ -z_{6n}\sin \left(\alpha +{\gamma }_n+{\eta }_n\right)-c_{1n}\sin \left(\alpha +{\gamma }_n\right)+b_{2n}\cos \alpha \\ \left\{z_{6n}\cos \left(\alpha +{\gamma }_n+{\eta }_n\right)+c_{1n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\cos \beta -a_{1n}\sin \beta \\ 1\end{array}]\end{array}& \left(27\right)\end{array}$ $\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{arm}.n}^r={}_{4n}{}^{r}T_{10n}^{4n}{\mathrm{TP}}_{\mathrm{armn}}^{10n}\\ =\left[\begin{array}{cccc}\cos \beta & \sin \beta \sin \left(\alpha +{\gamma }_n\right)& \sin \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& b_{2n}\sin \beta \sin \alpha +a_{1n}\cos \beta \\ 0& \cos \left(\alpha +{\gamma }_n\right)& -\sin \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \alpha \\ -\sin \beta & \cos \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \cos \beta \cos \left(\alpha +{\gamma }_n\right)& z_0+b_{2n}\cos \mathrm{\beta sin}\alpha -a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]·\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\theta }_n& -\sin {\theta }_n& e_{3n}\cos {\theta }_n\\ 0& \sin {\theta }_n& \cos {\theta }_n& c_{2n}+e_{1n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ =[\begin{array}{c}\left\{e_{1n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\sin \beta +a_{1n}\cos \beta \\ e_{1n}\cos \left(\alpha +{\gamma }_n+{\theta }_n\right)-c_{2n}\sin \left(\alpha +{\gamma }_n\right)+b_{2n}\cos \alpha \\ \left\{e_{1n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\cos \beta -a_{1n}\sin \beta \\ 1\end{array}]\end{array}& \left(28\right)\end{array}& \end{array}$ $\begin{array}{cc}\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{wheel}.n}^r={}_{4n}{}^{r}T_{12n}^{4n}{\mathrm{TP}}_{\mathrm{wheel}.n}^{12n}\\ =\left[\begin{array}{cccc}\cos \beta & \sin \beta \sin \left(\alpha +{\gamma }_n\right)& \sin \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& b_{2n}\sin \beta \sin \alpha +a_{1n}\cos \beta \\ 0& \cos \left(\alpha +{\gamma }_n\right)& -\sin \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \alpha \\ -\sin \beta & \cos \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \cos \beta \cos \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \mathrm{\beta sin}\alpha -a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \left({\theta }_n+{\zeta }_n\right)& -\sin \left({\theta }_n+{\zeta }_n\right)& e_{3n}\cos {\theta }_n\\ 0& \sin \left({\theta }_n+{\zeta }_n\right)& \cos \left({\theta }_n+{\zeta }_n\right)& c_{2n}+e_{3n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ =[\begin{array}{c}\left\{e_{3n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\sin \beta +a_{1n}\cos \beta \\ e_{3n}\cos \left(\alpha +{\gamma }_n+{\theta }_n\right)-c_{2n}\sin \left(\alpha +{\gamma }_n\right)+b_{2n}\cos \alpha \\ \{z_0+\left\{e_{3n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\cos \beta -a_{1n}\sin \beta \\ 1\end{array}]\end{array}& \left(29\right)\end{array}& \end{array}$ $\begin{array}{cc}\begin{array}{c}{P}_{\mathrm{touchpoint}.n}^r={}_{4n}{}^{r}T_{12n}^{4n}{T}_{13n}^{12n}{\mathrm{TP}}_{\mathrm{touchpoint}.n}^{13n}\\ =\left[\begin{array}{cccc}\cos \beta & \sin \beta \sin \left(\alpha +{\gamma }_n\right)& \sin \mathrm{\beta cos}\left(\alpha +{\gamma }_n\right)& b_{2n}\sin \beta \sin \alpha +a_{1n}\cos \beta \\ 0& \cos \left(\alpha +{\gamma }_n\right)& -\sin \left(\alpha +{\gamma }_n\right)& b_{2n}\cos \alpha \\ -\sin \beta & \cos \mathrm{\beta sin}\left(\alpha +{\gamma }_n\right)& \cos \beta \cos \left(\alpha +{\gamma }_n\right)& z_0+b_{2n}\cos \mathrm{\beta sin}\alpha -a_{1n}\sin \beta \\ 0& 0& 0& 1\end{array}\right]\\ \left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \left({\theta }_n+{\zeta }_n\right)& -\sin \left({\theta }_n+{\zeta }_n\right)& e_{3n}\cos {\theta }_n\\ 0& \sin \left({\theta }_n+{\zeta }_n\right)& \cos \left({\theta }_n+{\zeta }_n\right)& c_{2n}+e_{3n}\sin {\theta }_n\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z_{12n}\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]\\ =[\begin{array}{c}\left\{z_{12n}\cos \alpha +e_{3n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\sin \beta +a_{1n}\cos \beta \\ -z_{12n}\sin \alpha +e_{3n}\cos \left(\alpha +{\gamma }_n+{\theta }_n\right)-C_{2n}\sin \left(\alpha +{\gamma }_n\right)+b_{2n}\cos \alpha \\ z_0+\left\{z_{12n}\cos \alpha +e_{3n}\sin \left(\alpha +{\gamma }_n+{\theta }_n\right)+c_{2n}\cos \left(\alpha +{\gamma }_n\right)+b_{2n}\sin \alpha \right\}\cos \beta -a_{1n}\sin \beta \\ 1\end{array}]\end{array}& \left(30\right)\end{array}$