Auto-Tuning parameters of motion cueing algorithms for high performance driving simulator based on Kuka Robocoaster

Abstract

Driving simulators have been utilized to test and evaluate products and services for a long time. Their complexity and price range from extremely simple low-cost simulators with a fixed base to very complex high-end and pricey six-degree-of-freedom simulators with the XY table. The recent novel technique that uses an industrial robot - KUKA Robocoaster - as an interactive motion simulator platform, allowing for a highly flexible workspace as well as significantly lower prices due to mass production of the fundamental mechanics. In the constrained workspace of driving simulators, motion cueing algorithms (MCAs) are commonly employed to merge the tilt gravity and translational acceleration components for simulating the linear acceleration in the real vehicle. However, there is a few MCAs developed for the motion platform, almost MCAs were implemented for the standard six-degree-of-freedom simulators in the Cartesian coordinate. The classical MCA in the cylindrical coordinate (ClCy) MCA was first developed for the novel motion platform to take advantage of enormous rotational motion to simulate lateral acceleration while compensating for the bothersome longitudinal acceleration (due to centrifugal acceleration appearing in the rotational motion) with a proper pitch tilted angle. The process of tuning MCAs for the novel motion platform is time-consuming due to both trial and error method and the disturbing motion cues generated by rotational motion, thus it needs the involvement of experts. Although there are several auto-tuning approaches for classical, optimal, and model-predictive control MCA based on fuzzy control theory or genetic optimization method, the methods were purely applied for Cartersian coordinate without taking the bothersome longitudinal acceleration into account. Therefore, this paper firstly presents the process of integrating MCAs in the novel motion platform utilizing rotational motion for simulating lateral acceleration. For the case, besides the ClCy algorithm, the classical algorithm developed for the standart six-degree-of-freedom simulators was a sample implementation due to its popular and familiar characteristics. Secondly, the proposal of the use of the mean-variance mapping optimization (MVMO) for auto-tuning parameters of the two algorithms for reducing both rotational false cues in roll and pitch channel, and longitudinal acceleration as well as washout effect. The simulation results prove that 1) The classical and other MCAs can be applied in the novel motion platform with the proposed motion conversion; 2) both algorithms with auto-tuned parameters have high performance in exploiting effectively the workspace of the motion platform, producing no false cues of angular velocity, conpensating the disturbed longitudinal acceleration, and pulling the motion platform to the initial position after the simulation task; 3) The auto-tuning method is so transparent that can manipulates the specific simulated quantities according to the tuning goals.

Introduction

Driving simulation systems are a helpful and cheap technology that may be used for both instructional and research purposes. To provide the supporting motion, the systems originally employed a parallel hexapod robot called Stewart platforms which have a restricted area and relatively expensive prices. ¹ New motion platforms based on serial robots KUKA Robocoaster have recently been created to reduce these expenses and, most importantly, to increase the motion workspace, therefore expanding the application of the simulating process^2,3

The high fidelity level of simulated motion is ideally archived if the actual motion cues are totally exactly duplicated by the motion platform. However, the ideal motion cues are impossible to produce due to the motion platform's limited workspace. Therefore, motion cueing algorithms (MCAs) generate the scaled linear motion for the filtered linear high-frequency motion cues for by motion platforms and compensate the washout and disturbing cues by the suitable tilted angles. Concretely, the realistic acceleration is conventionally divided into two parts by MCAs: high-frequency and low-frequency. The tilt coordination approach simulates the low-frequency (LF) elements of acceleration, while the motion platform's transient linear acceleration reproduces the high-frequency (HF) components. The principle of the tilt coordination technique is to tilt the motion platform at a suitable angle to treat the human vestibular system. In this technique, the corresponding rotational rates are normally reduced to the perception threshold of the human vestibular system, so that the false cues of rotational motion are not perceived by the driver inside the simulator. The MCAs were specifically designed for various mechanical structures with reduced DOFs: the 2 DOF low-cost platform,^4–6 3 DOF VTI simulator, ⁷ and 5 DOF motion platform; ⁸ with the 6 DOF: KUKA Robocoaster, ⁹ the 6 DOF Stewart platforms, ¹ and 6 DOF Desdemona simulator in TNO; ¹⁰ with support axes as the 8 DOF driving simulator of the University of Leeds, ¹¹ and for 8 DOF including KUKA Robotcoaster and an XY-rail, recently. ¹²

There are various kinds of MCAs combining tilted angles and appropriate motion acceleration to simulate target motion cues. The traditional washout filter proposed by Conrad and Schmidt (1970) divided a cross-coupled linear low-pass filter (LP) and a linear high-pass filter (HP). ¹³ For the novel motion platform, Giordano et al. (2010) developed a new motion cueing algorithm named “cylindrical classical algorithm” to exploit the workspace of serial robots such as KUKA Robocoaster. ⁹ This algorithm is the classical washout algorithm that uses rotational end-effector motion for lateral acceleration instead of linear motion, yielding a larger acceleration workspace for serial robots. The “adaptive washout filter”^3,14–16 is a further modification of the conventional washout filter. In these methods, the washout filters’ parameters are systematically changed in real-time to optimize a cost function using the steepest descent techniques.

Later, Sivan et al. ¹⁷ created an optimum MCA, which was then refined by Reid and Nahon ³ and Telban et al. ¹⁸ The higher-order dynamic model of the vestibular system was later adopted by the authors. To make the simulated values match the target ones, the method employed higher-order washout filters rather than a classical and adaptive approach. Another offline optimum algorithm developed by Richard Romano ¹⁹ and Zywiol ²⁰ focused on determining the best mix of linear acceleration and tilt angle to imitate target motion signals. Besides, a unique optimum technology – model predictive control (MPC) – has recently been applied to address the problem of finding the right motion under the restriction of workspace boundaries and motion perception threshold.^12,21,22 According to the literature, the objective tuning parameters is to identify the suitable parameters, with which the MCAs create the simulated quantities within the required ranges or objective criteria such as performance indicator, ²³ Indicators for Reference Tracking Performance: Root Mean Square Error; Correlation Coefficient; Estimated Delay; and Indicators for Workspace Use: Interquartile range (IQR). ²⁴ Due to trial and error, the tuning process takes a long time and needs the assistance of specialists. Furthermore, Several auto-tuning techniques for various algorithms, including classical algorithms, optimum, and model-predictive control, based on the genetic approach have recently been used to discover the best values for parameters.^25–27

Note that most of all MCAs were developed for the Cartesian coordinate system motion platform, thus, the algorithms cannot be applied to every type of motion system. Besides, parameters of the algorithms are not clear physical meaning to be tuned for exploiting the flexibility of the novel motion platform. Therefore, this work focuses on 1) proposing an integration process for the MCAs, classical algorithm proposed by Reid and Nahon ³ as an example, into the motion platform based on KUKA Robocoaster; 2) proposing an auto-tuning process with mean-variance mapping optimization methods to improve the high performance of novel motion platform: tracking scaled motion cues, exploiting the workspace, and sastifying perception fidelity criteria such as removing rotational false cues in both roll and pitch channel, compensating the longitudinal acceleration, producing suitable scaled motion cues, and return to the initial position at the end of the simulation (washout-filter effect).

The remainder of the paper is organized in the following manner. Firstly, the coordinate systems are defined in the motion platform based on KUKA Robocoaster and the problems of integrating MCAs are presented in section 2. Secondly, the computation of trajectory generation for the end effector of the robot is introduced in section 3. Thirdly, the auto-tuning parameters process for CLRN and ClCy is described in section 4. The simulation results are discussed in the conclusion section 5.

Introduction of motion platform based on KUKA Robocoaster and the referecen frames

The motion platforms based on KUKA Robocoaster are used in several German organizations such as Max Planck Institute with MPI CyberMotion simulator ²⁸ and in the Laboratory for Mechanics and Robotics (LMR) at ‘The University of Duisburg- Essen ²⁹ (in Figure 1) with the specification in the Table 1. The motion platform consists of a KUKA KR500/1 TÜV Robocoaster robot with a Maurer Söhne rollercoaster seat mounted at the robot flange, a Visette 45 head mounted display unit, an A.R.T. The virtual reality environment is smoothly generated by the 3D engine and visualized in stereo 3D on the head-mounted display. A matching the trajectory is simultaneously provided to the motion platform as a function of time from the datasets delivered (offline) by the motion cueing algorithms. The input trajectory file contains the relative poses of the according to the sample time with time-step of 12.5 ms.

Figure 1.

The motion platform based on KUKA robocoaster.

Table 1.

Specification of the KUKA robocoaster.

Joint No.	Maximum angular acceleration ( $rad/s^2)$	Maximum angular velocity (deg / s)	Joint Limits
			Lower (deg)	Higher (deg)
1	1.70864	69	−167	167
2	1.26040	69	−96	−15
3	2.23308	69	−28	−17
4	0.73665	77	−350	−350
5	1.66237	76	−57.5	57.5
6	1.16660	120	−350	350

 Figure 2(a) shows the location of the reference frames and their relative position. Here, the inertial reference frame $K_0$ is at the base of the robot, the simulation frame K_S is at a specific position in the motion platform, the vestibular frame K_Ps is assumed at the pilot's vestibular system, and the end effector frame K_E is defined in the KUKA Robocoaster. Note that, for the investigated MCAs in this paper, the frame K_S and K_Ps are coincident ( $K_S\equiv K_{PS}$ ). The description of the frames is presented in detail as follow

Figure 2.

The position of KUKA robocoaster reference frames.

The inertial reference frame $K_0$ is defined in the robot's controller to make the seat-mounted at the end effector parallel to the ground. The frame has its origin located at a selected point in the first joint of the robot with a vertically upward $Z_0$ axis; $X_0$ axis points forward and $Y_0$ axis points to the left-hand side with respect to a driver. Note that an initial pose can be selected according to specific simulation tasks. In the paper, the initial pose is selected for a simulation task that has symmetric lateral motions which require the symmetric rotation of joint $q_1^R$ . Therefore, the position is corresponding robot's joint angles $q_i^R$ with $i\in \{1\cdots 6\}$ expressed as

$$[q_1^R,q_2^R,q_3^R,q_4^R,q_5^R,q_6^R]=[0,-21.37,-24.26,0,45.63,0].$$ (1)

Reference frame KE of End Effector has its origin at the End Effector of the KUKA Robocoaster. The $X_E$ axis is vertical downward, the $Z_E$ axis points forward and the $Y_E$ points to the left-hand side. From the initial position of the robot, the values of ${}_0^0{\underset{\_}{r}}_{E_0}$ is obtained

$${}_0^0{\underset{\_}{r}}_{E_0}=(x_{E_0},y_{E_0},z_{E_0})=(2.678,0,2.292)(m).$$ (2)

With ${}_A^C{\underset{\_}{r}}_B$ represents a relative position vector of a point B with regarding (w.r.t) a point A decomposed in the frame $K_C$ .

Reference frame $K_{Ps}$ is attached to driver's head hypothesized origin at driver's head. $X_{Ps}$ axis is parallel to the $X_S$ axis, and $Y_{Ps}$ axis is parallel to the $Y_{Ps}$ axis. The location of the frame $K_{Ps}$ is decomposed in frame $K_E$ as

$${}_E^E{\underset{\_}{r}}_{Ps}=(-a_E,0,b_E)$$ (3)

$a_E$ and $b_E$ are depended on drivers’ height. In the work, these values, $a_E\approx 0.55(m)$ and $b_E\approx 0.35(m)$ , were selected for the drivers with a medium height of 1.7 (m). The assumption can be applied for another driver if the rotational is not abruptly changed.

In oder to generate the relative pose of the motion platform, the relative position and orientation of the frame K_E w.r.t its initial position are firstly computed, then the inverted kinematic computation are applied. Therefore, several reference frames are defined as follow

The reference frame $K_1$ is obtained by rotating the frame $K_0$ around the $Z_0$ axis at an angle $\alpha$ . The role of the moving frame $K_1$ is to track the rotational motion around the $Z_0$ axis of the $K_{Ps}$ when $K_{Ps}$ moves along a planar circular trajectory.

The reference frame $K_2$ , $K_3$ are the initial positions of the frame $K_{Ps}$ and $K_E$ , respectively. Therefore, the position of these frames in the initial state, $t=0$ , (Figure 2(b)) are:

$$K_1\equiv K_0,K_2\equiv K_{Ps},\mathrm{and}{K}_3\equiv K_E.$$ (4)

For example, if the frame $K_{Ps}$ moves along a planar circular trajectory with the radius R and a tangent acceleration ${}^{Ps}{a}_{Ps}=[0,a_{Psy},0]$ (see Figure 2(b)), the frame $K_0$ rotates an angle $\alpha$ with the angular acceleration $\ddot{\alpha }$ to the frame $K_1$ and the frame $K_E$ moves also along a planar circular trajectory. Note that, if the frame $K_{Ps}$ has both the circular movement and a rotational one around the $X_{Ps}$ axis, the frame $K_E$ moves on a 3D curve.

Trajectory generation of KUKA Robocoaster

As mentioned above, the desired motion from an MCA, the classical MCAs as an example in Figure 3, divides target acceleration $a_{Vy}$ into the fast and slow part (sustained motion) which can be reproduced by the translational motion and tilted angle of the motion platform - using the tilt coordination block, respectively. In the part, the trajectory reproducing lateral acceleration is considered. Normally, the KUKA Robocoaster can work as a conventional motion platform such as a linear translational motion used to simulate lateral acceleration. Although, the usage makes the computation of the trajectory simple, the workspace of the robot is inefficiently used. Therefore, a circular movement, that exploits large rotational motion of the KUKA Robocoaster in the block “Robot trajectory generation”, is used to simulate the lateral acceleration instead of translational movement (Figure 3). As a result, the lateral motion of frame $K_S$ that was generated by a “MCA” block must firstly be converted into the rotational motion around the $Z_0$ axis, then the trajectory of the frame $K_E$ is computed accordingly. Note that, the circular trajectory causes a centrifugal acceleration that can be compensated by a tilt angle ${\theta }_c$ computed with the tilt coordination technique. As a result, the frame $K_S$ has three rotational motions including tilted roll angle around $X_S$ -axis, compensated pitch angle around $Y_S$ axis, and the circular movement $\alpha$ around $Z_0$ -axis.

Figure 3.

The diagram of robot trajectory generation for simulation of the pure lateral acceleration.

The block “Robot trajectory generation” receives the planned trajectory of the simulation frame $K_S$ to compute the corresponding trajectory of the frame $K_E$ of the KUKA Robocoaster by solving the problem: “Given orientation ${\underset{\_}{\beta }}_S$ and lateral acceleration ${\underset{\_}{a}}_S=[0,a_{Sy},0{]}^T$ of the frame $K_S$ , as well as the initial spatial position of the frame $K_S$ and $K_E$ , compute the relative position and orientation of the frame $K_E$ w.r.t the frame $K_3$ in order to make frame $K_S$ move along a planar circular curve around $Z_0$ axis. Additionally, the tangent acceleration of the rotation movement equals the lateral acceleration $a_{Sy}$ .” Hence, ${}_3{\underset{\_}{\beta }}_E$ and ${}_3^3{\underset{\_}{r}}_E$ are the variables are found (see Figure 3), then, relative position $\mathrm{\Delta }{\underset{\_}{r}}_E{=}_3^3{\underset{\_}{r}}_E$ , the orientation $\mathrm{\Delta }{\underset{\_}{\beta }}_E{=}_3{\underset{\_}{\beta }}_E$ , and the incremental values of sample time $\mathrm{\Delta }t$ are used to build the “Robot Data” that stores the incremental position of the End Effector.

Figure 4 shows the process of computing the trajectory for the KUKA Robocoaster in detail. The input data entering into the “compute a circular trajectory” block includes the simulated quantities created by a washout algorithm MCA, relative spatial relationship of the reference frames $K_0,K_2,K_3$ . The simulated quantities comprise of motion quantities of the frame $K_S$ in the cabin, the lateral component of acceleration $a_{Sy}$ , and relative orientation ${\underset{\_}{\beta }}_S$ w.r.t the frame $K_0$ .

Figure 4.

The diagram of the computing the circular motion.

The “compute a circular motion” block, that bases on the principle of the cylindrical classical MCA, ⁹ converts the linear lateral trajectory of the frame $K_{Ps}$ to the circular one (see dash line in Figure 2(b)). In the block, the given variables are in the blue solid boxes, while the solution (found variables) are in the red dash boxes. Besides, the magenta solid box represented the mathematical operation and transformation. Note that, there are two sub-processes (bounded box with black dash line) in the block, that are used to compute the relative position and orientation of the frame $K_E$ w.r.t the frame $K_3$ , respectively. The “Cen Accel Computation” and “Tilt Coord” blocks are used to compute the compensated pitch angle ${\theta }_c$ . Note that the “Selector” block is used to decide whether the centrifugal acceleration is compensated or not. The computating processes are described in detail as follow.

The relative position of End Effector ${}_3^3{\underset{\mathbf{\_}}{\mathit{r}}}_{\mathit{E}}$ is the position vector of the frame $K_E$ w.r.t the frame $K_3$ is decomposed in the frame $K_3$

$${}_3^3{\underset{\_}{r}}_E{=}^3{\mathbf{R}}_2{}_3^2{\underset{\_}{r}}_E.$$ (5)

Where

$${}^3{\mathbf{R}}_2^T={}^2{\mathbf{R}}_3={\mathbf{R}}_{Y_2}(\pi /2)=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right].$$ (6)

and

$${}_3^2{\underset{\_}{r}}_E={}_3^2{\underset{\_}{r}}_2+{}_3^2{\underset{\_}{r}}_{Ps}+{}_{Ps}^2{\underset{\_}{r}}_E$$ (7)

$${=}_3^2{\underset{\_}{r}}_2{+}_2^2{\underset{\_}{r}}_{Ps}+{}^2{\mathbf{R}}_{Ps}{}_{Ps}^{Ps}{\underset{\_}{r}}_E.$$ (8)

Because the frame $K_{Ps}$ is parallel to the frame $K_E$ and there is no relative motion between two frames. Therefore,

$${}_3^2{\underset{\_}{r}}_2={-}_{Ps}^{Ps}{\underset{\_}{r}}_E=-{}_2^2{\underset{\_}{r}}_3=[-b_E,0,-a_E{]}^T.$$ (9)

Besides

$${}_2^2{\underset{\_}{r}}_{Ps}={}_0^2{\underset{\_}{r}}_{Ps}-{}_0^2{\underset{\_}{r}}_2$$ (10)

Because the frame $K_0$ is parallel with the frame $K_2$ , the rotational matrix ${}^0{\mathbf{R}}_2=\mathbf{I}$ . Thus

$${}_2^2{\underset{\_}{r}}_{Ps}={}_2^0{\underset{\_}{r}}_{Ps}={}_0^0{\underset{\_}{r}}_{Ps}-{}_0^0{\underset{\_}{r}}_2.$$ (11)

By using the “compute a circular motion” block, the frame $K_{Ps}$ moves along a planar circular curve with radius R and an angular acceleration $\ddot{\alpha }$ that makes a tangent acceleration equivalent to the lateral acceleration. Therefore,

$$R\ddot{\alpha }=a_{Psy}.$$ (12)

where the radius of the circular movement is

$$R=\sqrt{x_{E_0}^2+y_{E_0}^2}+b_E.$$ (13)

Because the frame $K_{Ps}$ moves along a circular trajectory, therefore, the position vector of the frame $K_{Ps}$ is decomposed in the frame $K_0$ as

$${}_0^0{\underset{\_}{r}}_{Ps}=[R\cos \alpha ,R\sin \alpha ,z_{E_0}+a_E{]}^T.$$ (14)

and the location of the frame $K_2$ is

$${}_0^0{\underset{\_}{r}}_2=[R\cos \alpha ,R\sin \alpha ,z_{E_0}+a_E{]}^T.$$ (15)

Therefore

$${}_2^2{\underset{\_}{r}}_{Ps}=[R(\cos \alpha -1),R\sin \alpha ,0{]}^T.$$ (16)

When the frame $K_{Ps}$ moves along a circular trajectory with a radius R and an angular velocity $\dot{\alpha }$ , the centrifugal acceleration $a_R$ acting on the frame $K_{Ps}$ is

$$a_R=R{\dot{\alpha }}^2.$$ (17)

An extra pitch tilted ${\theta }_c$ angle is generated by rotating the frame $K_{Ps}$ around its y-axis $y_{Ps}$ to compensate for the disturbance acceleration as the conventional tilt-coordination technique. Therefore,

$${\theta }_c=-\arcsin {\displaystyle \frac{a_R}{g}.}$$ (18)

Let ${}_1{\underset{\_}{\beta }}_{Ps}$ is the Euler angles of the frame $K_{Ps}$ w.r.t the frame $K_1$ ,

$${}_0{\underset{\_}{\beta }}_{Ps}={}_1{\underset{\_}{\beta }}_{Ps}+{}_0{\underset{\_}{\beta }}_1={}_1{\underset{\_}{\beta }}_{Ps}+[0,0,\alpha {]}^T.$$ (19)

Hence, besides a lateral acceleration, a roll angle that was generated from an MCA, there is a pitch angle that is used to compensate for the centrifugal acceleration produced from the circular movement.

$${}_1{\underset{\_}{\beta }}_{Ps}=[{\phi }_S,{\theta }_c,0].$$ (20)

From eq. 19 the Euler angle of the frame $K_{Ps}$ with regard to $K_0$ is

$${}_0{\underset{\_}{\beta }}_{Ps}=[{\phi }_S,{\theta }_c,\alpha {]}^T.$$ (21)

Note that the frame $K_0$ is parallel to the frame $K_2$ , thus ${}_0{\underset{\_}{\beta }}_{Ps}{=}_2{\underset{\_}{\beta }}_{Ps}$ . From the Euler angle rotation matrix with three angles ${\phi }_S$ , ${\theta }_c$ , and $\alpha$ is described as

$${}^2{\mathbf{R}}_{Ps}=\left[\begin{array}{ccc}{R}_{11}^{*}& R_{12}^{*}& R_{13}^{*}\\ R_{21}^{*}& R_{22}^{*}& R_{23}^{*}\\ R_{31}^{*}& R_{32}^{*}& R_{33}^{*}\\ & & \end{array}\right]$$ (22)

The matrix can be expressed in detail such as

$${}^2{\mathbf{R}}_{Ps}=\left[\begin{array}{ccc}\cos \alpha \cos {\theta }_c& -\sin \alpha \cos {\phi }_S+\cos \alpha \sin {\theta }_c\sin {\phi }_S& \sin \alpha \sin {\phi }_S+\cos \alpha \sin {\theta }_c\cos {\phi }_S\\ \sin \alpha \cos {\theta }_c& \cos \alpha \cos {\phi }_S+\sin \alpha \sin {\theta }_c\sin {\phi }_S& -\cos \alpha \sin {\phi }_S+\sin \alpha \sin {\theta }_c\cos {\phi }_S\\ -\sin {\theta }_c& \cos {\theta }_c\sin {\phi }_S& \cos {\theta }_c\cos {\phi }_S\end{array}\right]$$ (23)

From the eq. 9, eq. 11, eq. 16 and eq. 23, the position ${}_3^2{\underset{\_}{r}}_E$ is archieved, then ${}_3^3{\underset{\_}{r}}_E$ the relative position of the frame K_E is $\mathrm{computed}\mathrm{as}\mathrm{in}\mathrm{eq}.5\mathrm{to}\mathrm{obtain}$

$${}_3^3{\underset{\_}{r}}_E=\left[\begin{array}{c}-b_E\sin {\theta }_c-a_E(1-\cos {\theta }_c\cos {\phi }_S)\\ R\sin \alpha -b_E\cos {\theta }_c\sin \alpha -a_E(-\cos \alpha \sin {\phi }_S+\sin \alpha \sin {\theta }_c\cos {\phi }_S)\\ -R(1-\cos \alpha )+b_E(1-\cos \alpha \cos {\theta }_c)-a_E(\sin \alpha \sin {\phi }_S+\cos \alpha \sin {\theta }_c\cos {\phi }_S)\end{array}\right]$$ (24)

The relative orientation of End Effector ${}_3{\underset{\mathbf{\_}}{\mathit{\beta }}}_{\mathit{E}}$ is vector of the Euler angles of the frame $K_E$ w.r.t the frame $K_3$ . The rotational matrix between the frame $K_E$ and the frame $K_3$ is

$${}^3{\mathbf{R}}_E{=}^3{\mathbf{R}}_2^2{\mathbf{R}}_{Ps}^{Ps}{\mathbf{R}}_E.$$ (25)

It is assumed that a driver is tightly fastened in a roller coaster seat equipped at the End Effector, therefore there is no relative motion between the two frames $K_E$ and $K_{Ps}$ . Thus, the given rotational matrices are:

$${}^E{\mathbf{R}}_{Ps}={\mathbf{R}}_{Y_{Ps}}\left(-{\displaystyle \frac{\pi }{2}}\right);{}^2{\mathbf{R}}_3={\mathbf{R}}_{Y_2}(\pi /2)=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ -1& 0& 0\end{array}\right].$$ (26)

Replacing the eq. 26 into the eq. 25, the rotational matrix is obtained such as

$${}^3{\mathbf{R}}_E=\left[\begin{array}{ccc}{R}_{33}^{*}& -R_{32}^{*}& -R_{31}^{*}\\ -R_{23}^{*}& R_{22}^{*}& R_{21}^{*}\\ -R_{13}^{*}& R_{12}^{*}& R_{11}^{*}\\ & & \end{array}\right].$$ (27)

Therefore, comparing the eq. 27 to eq. 22 with the Euler angle ${}_3{\underset{\_}{\beta }}_{\mathit{E}}$ replacing for the ${}_0{\underset{\_}{\beta }}_{Ps}$ (eq. 21), the following expression archived

$$\{\begin{array}{ccc}-\sin {\theta }_E& =& -R_{13}^{*}\\ \sin {\phi }_E\cos {\theta }_E& =& -R_{23}^{*}\\ \sin {\psi }_E\cos {\theta }_E& =& R_{12}^{*}\end{array}.$$ (27)

Solving these equations, the Euler angles of $K_E$ w.r.t $K_3$ are

$$\{\begin{array}{ccc}{\theta }_E& =& \arcsin (R_{13}^{*})\\ {\phi }_E& =& \arcsin \left({\displaystyle \frac{R_{12}^{*}}{\cos {\theta }_E}}\right)\\ {\psi }_E& =& \arcsin \left({\displaystyle \frac{-R_{23}^{*}}{\cos {\theta }_E}}\right)\end{array}.$$ (28)

From equation eq. 22, eq. 23 and eq. 28, the Euler angles are described as

$$\{\begin{array}{ccc}{\theta }_E& =& \arcsin (\sin \alpha \sin {\phi }_S+\cos \alpha \sin {\theta }_c\cos {\phi }_S)\\ {\phi }_E& =& \arcsin \left({\displaystyle \frac{\cos \alpha \sin {\phi }_S-\sin \alpha \sin {\theta }_c\cos {\phi }_S}{\cos {\theta }_E}}\right)\\ {\psi }_E& =& \arcsin \left({\displaystyle \frac{-\sin \alpha \cos {\phi }_S+\cos \alpha \sin {\theta }_c\sin {\phi }_S}{\cos {\theta }_E}}\right)\end{array}.$$ (29)

In conclusion, the relative spatial position of the frame $K_E$ are ${}_3^3{\underset{\_}{r}}_E=[x_E,y_E,z_E{]}^T$ , ${}_3{\underset{\_}{\beta }}_E=[{\phi }_E,{\theta }_E,{\psi }_E]$ .

Here, $x_E,y_E,z_E$ are from eq. 24 and ${\phi }_E,{\theta }_E,{\psi }_E$ are from eq. 29. The trajectory of the End of Effector of the Kuka Robocoaster is stored as a text file containing the relative position and orientation according to the sample time with time-step of 12.5 ms.

Objective criteria and auto-tuning process for MCAs

White and Rodchenko ³⁰ mentioned that the effect of the MCAs parameters on motion fidelity is still ambiguous. Thus, it is not a clear method for improving the fidelity by enhancing the benefit motion cues stimulating the vestibular system or combining motion cues from the multi-sensory system. In the literature, the tuning process for an MCA, which is an almost classical algorithm, is usually implemented in two steps. At first, the parameters are tuned to minimize the motion cues errors or use the Sinacori criterion, ³¹ then the achieved parameters are validated by the subjective test There are several works mentioning the relationship between the parameters of the classical algorithm with the motion fidelity. For instance, Grant and Reid ³² analyzed the effect of the washout filters’ parameter on the false cues and introduce a way to reduce the undesired cues by adjusting the scale factor k, break frequency of the washout filters, and rotational threshold. etc. Later, Grant et al., ³³ Bruenger-Koch, ³⁴ and Jamson ¹¹ implemented research on some set of parameters of the classical algorithm to assert that the suitable parameters provide better motion fidelity with reduced shape error, and increase scale errors. Although the tuning processes were not mentioned in detail, the response of the washout filters could be the result of reducing false cues by flattening the bode plot. From the review of the objective tuning method, the paper uses the objective criteria for the parameter tuning process by trial and error method ²⁷ and proposes an automatic tuning process that improve the performance of the MCA (Figure 5) as in the following parts.

Figure 5.

Tuning methods for a MCA.

Objective criteria and trial and error tuning process

Normally, for the objective tuning process, the parameters of the MCAs are tuned to meet every objective criterion defined by the workspace limitation and the constraints of the motion perception. In the paper, the objective prioritized criteria C_i, with i = 1 − 4 for the tuning process are described as following in detail:

• C₁: Limiting the simulator's movement and avoiding the vertical false cues by which there is none of the limit violation and the protection of the tilt-coordination technique.

• C₂: Avoiding the rotational false cues and unexpected false cues due to the tilt angular acceleration and avoiding the false cues of the specific force

• C₃: Select the suitable scale factors to avoid too weak cues.

• C₄: Exploiting the available workspace for linear simulated acceleration and pulling the motion platform to the initial position at the end of the simulation task.

Based on the 4 tuning criteria, the trial and error tuning process (Figure 5) considers C₁ as the most significant criterion, while C₄ is considered as the least significant one. The tuning process of a MCA firstly convert the desired linear motion to corresponding circular trajectory of the novel motion platform and compute the simulated quantities by the “Convert to Robocoaster Platform” block (the computation trajectory described in the section III in detail). Afterward, The simulated quantities can be checked with “ Tuning Criteria” block to generate the check result G_j that composes of all 4 tuning criteria C_i, with i = 1–4. If all tuning criteria are sastifies, the expexted parameter set is found, otherwise the parameters are adjusted according to the design perspective. The accepted parameter set could be not always sastified all the objective criteria.

Auto-tuning process for objective criteria with Mean variant mapping optimization method

The trial and error tuning process is time consuming and depends on the designer's experience with each of MCA, thus, a common auto-tuning process being able to applied any MCAs was developed with novel optimization method named Mean-variance mapping optimization (MVMO). The method developed by Erlich et al. ³⁵ is a member of the family of so-called population-based stochastic optimization techniques, which incorporates information on the performance of the specific number of best individuals to minimize a particular cost function. Comparing to other heuristic approach, the MVMO uses the special mapping function with variant mapping curve shape and location influenced by the seaching process. The MVMO has better convergence than other heuristic methods in IEEE 57 and 118- bus system test case in a comparative study for optimal reactive power dispatch with defined constraints. ¹⁵

In order to automatically tune the MCA parameters, an optimization can be run using the objective measure as a cost function of the workspace limits and motion perception cues. This is described in the following for the case CLRN.

For auto-tuning the cost function with regard to the criteria $C_i,i=\{1-4\}$ defined as

$$F_c=\sum _{k=1}^4{w}_k{J}_k$$ (30)

where $J_k$ and $w_k,\mathrm{with}k=\{1-4\}$ are the penalty functions and weighting values corresponding to the criteria $C_i,\mathrm{with}i=\{1-4\}$ . For pulling simulated quantities to the desired range, the exponent functions are applied when the simulated quantities are over their threshold values. The reason is that the exponent function produces high value when its variable has small change. For example, of the penalty function of angular velocity has the abrupt raise when the angular velocity increases above its maximum value ${\omega }_{max}$ (Figure 6). Therefore, the penalty functions are defined as

Figure 6.

Exponents penalty function and the effect of the penalty functions the angular velocity.

Workspace penalty function J₁

The position penalty function $J_1$ is the most important because it protects the simulated motions from violations of the hard limits of the driving simulator. Let $D_E$ store the position and orientation of the frame $K_E$ found in the section III, such as

$$D_E=\{D_{Ej}\},(j=1\cdots 6)=\{x_E,y_E,z_E,{\phi }_E,{\theta }_E,{\psi }_E\}\in \{L_j,U_j\}.$$ (31)

Here, $L_j$ and $U_j$ are the lower and upper limit values of the $D_{Ej}$ that define the available workspace for the frame $K_E$ (shown in the Table 2).

Table 2.

Position and orientation limits of the End of effector w.r.t to the intial pose.

	${}_3^{}{\underset{\_}{\beta }}_E^2$			${}_3^3{\underset{\_}{r}}_E^2$
${\phi }_E^2(rad)$	${\theta }_E^2(rad)$	${\psi }_E^2(rad)$	$x_E^2(m)$	$y_E^2(m)$	$z_E^2(m)$
$[-3.605,3.604]$	$[-0.13,0.32]$	$[-0.54,0.54]$	$[-0.1,0.27]$	$[-2.70,2.74]$	$[-5.43,0.073]$

Denote $D_{Ej,min}$ and $D_{Ej,max}$ as the minimum and maximum values of the $j^{th}$ dof in the trajectory, respectively. In addition, the minimum set $V_{min}=\{V_{j,min}\}{,}_{(j=1\cdots 6)}$ and maximum set $V_{max}=\{V_{j,max}\}{,}_{(j=1\cdots 6)}$ , which describe whether a DOF $j_{th}$ has a violation, are defined as

$$V_{j,min}=\{\begin{array}{cc}{L}_j& if{D}_{Ej,min}\ge L_j\mathrm{\&}{D}_{Ej,min}\le U_j\\ D_{Ej,min}& if{D}_{Ej,min}<L_j\end{array},$$ (32)

Here, the minimum set $V_{j,min}=L_j$ means that there a violation of the lower physical limit of the $j^{th}$ DOF, otherwise there is no violation. A similar rule is applied to the maximum set (eq. 4).

$$V_{j,max}=\{\begin{array}{cc}{U}_j& if{D}_{Ej,max}\le U_j\mathrm{\&}{D}_{Ej,max}\ge L_j\\ D_{Ej,max}& if{D}_{Ej,max}>U_j\\ & \end{array}.$$ (33)

The workspace penalty function $J_1$ is expressed as

$$J_1=\sum _{j\in \{1\cdots 6\}}(e^{{(V_{j,min}-L_j)}^2}-1)+(e^{{(V_{j,max}-U_j)}^2}-1).$$ (34)

If there is no violation of the lower and the upper physical limits, $J_1=0$ .

False cue penalty function J₂

This function manipulates three kinds of potential false cues which are angular velocity, acceleration over their threshold levels (Figure 6) and serious missing cues of the simulated specific force.

$$J_2=J_{\omega }+J_{\dot{\omega }}+J_f$$ (35)

The penalty functions are defined as follows

Angular velocity penalty function ${\mathit{J}}_{\mathit{\omega }}$

The rotational false cues due to the angular, ${\omega }_S$ can be eliminated by the penalty functions $J_{\omega }$ which removes the undesired quantities that are above the threshold values of the semicircular system ${\omega }_{th}$ ,.

$$J_{\omega }=\{\begin{array}{cc}0& if|\omega {|}_{max}<{\omega }_{th}\\ e^{{(|\omega {|}_{max}-{\delta }_S)}^2}-1& if|\omega {|}_{max}\ge {\omega }_{th}\end{array},$$ (36)

where $|\omega {|}_{max}$ is the maximum of the absolute simulated angular velocity, and the limit value ${\omega }_{th}$ is the angular velocity threshold of the semicircular system.

Angular acceleration penalty function ${\mathit{J}}_{\dot{\mathit{\omega }}}$

Similarly, the penalty function for angular acceleration is defined as

$$J_{\dot{\omega }}=\{\begin{array}{cc}0& if|\dot{\omega }{|}_{max}<{\dot{\omega }}_{th}\\ e^{{(|\dot{\omega }{|}_{max}-{\dot{\omega }}_{th})}^2}-1& if|\dot{\omega }{|}_{max}\ge {\dot{\omega }}_{th}\end{array},$$ (37)

where $|\dot{\omega }{|}_{max}$ is the maximum of the absolute simulated angular acceleration, and the limit values ${\dot{\omega }}_{th}$ is the angular acceleration threshold of the semicircular organ.

Specific forces penalty function ${\mathit{J}}_{\mathit{f}}$

The serious missing cues (distorting cues and negative cues shown in Figure 7) of the simulated specific forces can be numerically described by the square error of the simulated specific force $f_{Ps}$ regarding the scaled vehicle specific force $f_{Vy}$ as the equation

$$e_f=\sqrt{{(f_{Ps}-k_S{f}_{Vy})}^2}.$$ (38)

Figure 7.

Types of false cues specific force.

Here the simualted scale signal $k_S=\frac{\int f_{Ps}dt}{\int f_{Vy}dt}$ when the simulated specific force has good correlation to the target specific force.

There errors should be smaller than the threshold value of the otolith system ${\delta }_O$ as mentioned in, ²³ thus a penalty function $J_{fsh}$ can be defined as

$$J_f=\{\begin{array}{cc}0& if|e_f{|}_{max}<{\delta }_O\\ e^{{(|e_f{|}_{max}-{\delta }_O)}^2}-1& if|e_f{|}_{max}\ge {\delta }_O\end{array},$$ (39)

Here, the $|e_f{|}_{max}$ is the maximum of the absolute values of the error.

Scale factor penalty function ${\mathit{J}}_3$

The penalty function $J_{sc}$ pulling the factor to the expected values $k_{S,ex}$ can be defined as:

$$J_3=e^{{(k_S-k_{S,ex})}^2}-1;$$ (40)

Here, $k_{S,ex}\in [0.4,1]$ defines the range of the expected scale factors. The larger $k_{S,ex}$ is selected, the larger the scale factor can be found.

Exploiting workspace penalty functions ${\mathit{J}}_4$

Translational motion penalty function ${\mathit{J}}_{\mathit{t}\mathit{r}}$

The workspace of the motion platform can be effectively exploited if the ratio of simulated acceleration to the target acceleration is increased. Thus, the ratio $k_a=\frac{\int a_{S}dt}{\int a_{V}dt}$ is constrained by the lower limit $k_{a,min}$ to construct the penalty function $J_{tr}$ such as

$$J_{tr}=\{\begin{array}{cc}0& if{k}_a\ge k_{a,min}\\ e^{{(k_a-k_{a,min})}^2}-1& if{k}_a<k_{a,min}\\ & \end{array}.$$ (41)

Washout penalty function ${\mathit{J}}_{\mathit{W}\mathit{o}}$

Considering the washout effect of an algorithm, the penalty function $J_W$ is defined to pull the angles of the simulator to zero at the end of the simulation $t_{End}$ (i.e ${\phi }_S(t_{f,i})\to 0$ , with $t_{f,i}\in \{t_{End}-N_f*{\mathrm{\Delta }}_T,t_{End}\}$ )

$$J_W=e^{{\left(\sum _{i\in \{1\cdots N_f\}}g\cdot {\phi }_S(t_{f,i})\right)}^2}-1,$$ (42)

By using the penalty functions mentioned above, the constrained conditions are taken into account during the tuning process.

Result and discussion

The method of calculating the robot's motion trajectory is successfully applied to both algorithms, CLRN and ClCy corresponding to Casterian and Cylindrical-coordinate systems, to implement the $C_1$ criterion - check whether the simulated motions generated by the algorithms are in the available operation workspace of the robot or not (the limited workspace shown in Table 2). This check is a crucial part of the parameter tuning process.

The parameters of two MCAs algorithms are tuned both with trial and error and with auto-tuning method for the simulation case that is the pure lateral acceleration of a roller coaster moving with constant velocity on a flat S-shaped track (the scaled of lateral acceleration shown by solid black line in Figure 8(a)). For the trial and errors tuning method (Figure 5), first the simulated quantities are used to compute the objective criteria C₁₋₄ of motion perception that relate to the specific factors as the simulated scale factor k_S, ratio of translational motion k_a, shape errors of specific forces e_f, and the restricted angular velocity and acceleration with limited values in Table 3. Because the parameters of the MCA are not transparent to the objective criteria, the noticed changes of the objective criteria and the simulated quantities for every trials are observed and recorded for the next trials. The tuning process can stop when all the objective criteria reach to its desired values. For the first tuning parameters with trial and error method, it consumes neary two hours for each algorithm for fine adjusting each parameters.

Figure 8.

Responses of the CLRN algorithm with trial and error-tuned parameters.

Table 3.

Selected threshold and limited range for tuning quantities.

${\delta }_o$	${\omega }_{th}$	${\dot{\omega }}_{th}$	$k_{S,min}$	$k_{S,max}$	$k_a$
$0.17(m/s^2)$	$6(deg/s)$	$11(deg/s)$	$0.38$	$1$	$0.50$

Figure 8 and Figure 10 depict the translational and rotational motion of the simulation platform with the trial and error parameters of the CLRN and ClCy algorithms. In the simulation result, the target lateral specific forces are scale down with the with the scaling factor $k=0.4$ for the transparent comparison with the simulated specific forces. It can be seen that the scaled target specific force is fully reconstructed in both magnitude and phase by the simulated accelerations and compensatory accelerations based on the suitable tilted angle of the simulation platform (Figures 8a, 10a shows the maximum shape errors factor $e_f^{*}$ insignificant). In addition, for the washout effect (one factor in the criteria C₄), the simulated accelerations and tilted are pulled to the initial position at the end of the simulation cycle.

Figure 10.

Responses of the clCy algorithm with trial and error-tuned parameters.

In addition, Figure 8(b) and Figure 10(b) depict the simulated angular velocities and angular accelerations both below the perceived rotational motion (the factors in the criteria C₂). Note that, although only lateral acceleration is simulated, the algorithms generate pitch angles to compensate for the centrifugal acceleration caused by the circular motion of the robot. The CLRN algorithm produces smaller roll velocities and accelerations but larger pitch velocities and accelerations than the CyCl algorithm. This is because the CLRN algorithm with $k_a$  = 0.73 generates langer circular motions than the CyCl algorithm with ( $k_a$  = 0.43).

By applying the parameter found by auto-tuning method with MVMO for CLRN and ClCy, the limit values for the simulated quantities used for the penalty functions are list in the Table. 2 and Table 3. Besides, the weights of cost function terms are selected according to the priority levels (C₁ is the most important) of the objective criterial (Table 4). The priority levels presenting by the values of weights guide the auto-tuning method to provides the parameter sets with which the simulated quantities can be implemented in the motion platform with desired motion fidelity. The selection of the weighs is easy to approach due to its transparency related to the objective criteria/penalty functions. The undesired behaviour of the simulated motion can easily be solved by fine tuning weights according to the penalty functions related to behaviour. For example, if the simulated angular velocity is above its threshold values, the weight $w{J}_{\omega }$ must be increased. The auto-tuning process for each MCA in the simulation task has a quick convergence and consumes nearly 5 min running in the HP Zbook G3 for 300 test samples with a couple of time for fine tuning weights.

Table 4.

Selected weighting values of the penalty functions for CLRN and clCy algorithms.

Penalty functions	$w{J}_p$	$w{J}_{\omega }$	$w{J}_{\dot{\omega }}$	$w{J}_{sc}$	$w{J}_{tr}$	$w{J}_{f_{sh}}$	$w{J}_{W_o}$
Weighting values (CLRN)	$2^{12}$	$2^6$	$2^3$	$2^2$	$1$	$2^4$	$2^1$
Weighting values (ClCy)	$2^{10}$	$2^4$	$2^3$	$2^2$	$1$	$2^4$	$2^1$

The simulation results (Figure 9 and Figure 11) show that the simulated specific force of both CLRN and ClCy algorithm not noly tracking the scale targets specific force with larger scale factors but also preventing the false cues of motion. In concretely, the ratio of the circular motion of CLRN has been adjusted down ( $k_a=0.5$ ) to increase the tilted motion while keeping get the simulated angular velocities and accelerations below the allowable threshold. Note that, the motion platform with auto-tuned parameters is pulled to the original position earlier than one with trial and errors approach. Especitally, the auto-tuned parameters employ better workspace to increase the simulated scale factor $k_S=0.41$ than in the case of using the trial and error parameter ( $k_S=0.39$ ). On the other hand, for the ClCy algorithm (Figure 11(a)), the ratio of translation motion increases ( $k_a=0.5$ ) and the simulated scale factor ( $k_S=0.43$ ) compared to the case of using trial and error parameters (Figure 9(a)). Furthermore, the angular velocity and acceleration of roll and pitch channels are unders its theshold values (Figure 11(b)). Besides, the ClCy with auto-tuned parameters generates larger rotational motion than the algorithm with trial and error parameters (Figure 9(b)).

Figure 9.

Responses of the CLRN algorithm with auto-tuned parameters.

Figure 11.

Responses of the clCy algorithm with auto-tuned parameters.

Conclusion

This paper presents a method to integrate MCAs algorithms built on a cartesian coordinate system into a driving simulation system using KUKA Robocoaster. This integrated approach exploits the KUKA Robotcoaster's ability to use circular motions to reproduce high-frequency lateral accelerations while eliminating false cues caused by centrifugal acceleration. Next, the method of parameter auto-tuning of the algorithms is proposed to adjust the parameters of the CLRN and ClCy algorithms with the criteria of workspace limitation and exploitation, and false cues restriction for both roll and pitch channel, as well as improve the washout effect. The test results of applying the two algorithms, CLRN and ClCy, to the roller coaster simulation with only lateral acceleration show that the CLRN algorithm can be applied offline to the KUKA Robotcoaster using the appropriate robot trajectory calculation steps. The parameter auto-tuning method with objective functions related to mentioned tuning criteria can find the right set of parameters for the effective use of the motion platform's workspace while ensuring the elimination of false cues including the dirturbing cues due to the rotational trajectory of motion platform. The auto-tuning method provides the transparent method for specific performance criteria and has the quick convergence with small test samples. Thus, the method save the time consuming comparing with the trial and error approach. In addition, this method is also applicable to different algorithms based on the common objective function in the process of finding suitable parameters. In particular, the washout effect objective function in the cost function of the auto-tuning method forces the simulation platform to return to its initial state. The washout effect normally consumes tuning time by trial and error. The preliminary results of auto-tuning method prove the applicability of the method to other algorithms with other tuning criteria suitable for specific simulation goals in the future.

Author biographies

Duc-An Pham is a doctor in the School of Mechanical Engineering, Hanoi University of Science and Technology, Vietnam. He got a Bachelor degree in the Talented Engineering Programme of Mechatronics and a Master degree in Mechanics Engineering at Hanoi University of Science and Technology, Vietnam. Then he graduated PhD degree in Mechatronics and Robotics at Duisburg-Essen University, Germany. His research interests are Machining Process, CAD/CAM/CAE, Robotics, Motion cueing algorithm, Driving simulators and Auto-tuning method. Currently, he serves as a Vice-Dean of Mechatronics Faculty.

Duc-Toan Nguyen is a professor in the School of Mechanical Engineering, Hanoi University of Science and Technology (HUST), Vietnam. He was educated at the Hanoi University of Technology for his Engineer and Master's degree in the area of Mechanical Engineering and graduated PhD at the Kyungpook National University. Then, he pursued his professional career at the leading university of Vietnam (HUST), where he undertook research related to the Plasticity of Material, Machining Processes, CAD/CAM/CAE, and Robotics. Currently, he serves as the Vice-president of the Vietnam Association of Science Editing (VASE).