A Design and an Implementation of an Inverse Kinematics Computation in Robotics Using Gröbner Bases

In robotics, inverse kinematics computation is one of the central topics in motion planning [17]. In the field of computer algebra, methods of the inverse kinematics computation using Gröbner bases have been proposed ([2, 7, 22, 23], and the references therein). After formulating the forward kinematics problem using the Denavit–Hartenberg convention, the inverse kinematics problem is derived as a system of algebraic equations by conversion of trigonometric expressions to polynomials. Then, the system is triangularized by computing the Gröbner basis with respect to the lexicographic ordering and solved by appropriate solvers.

Since methods using Gröbner bases solve the inverse kinematics problem directly, these methods have advantages that one can verify if the given inverse kinematics problem is solvable (with the certification of the solution if needed), and if it is solvable, one can obtain the configuration of parameters for the desired motion of the robot directly, before the actual motion. On the other hand, the computational cost of methods using Gröbner bases tends to be high compared to that of numerical methods, thus it is desired to decrease computational cost for effective computation of solving the inverse kinematics problem using Gröbner bases. Furthermore, for the use of these methods in robotics simulators such as the Robot Operating System (ROS) [9], an implementation that can easily be integrated with these simulators is needed.

In this paper, we present the solution and a portable implementation of the inverse kinematics computation of a 3 degree-of-freedom (DOF) robot manipulator using Gröbner bases. For portable implementation and rapid development, the main program is developed in Python with computer algebra system SymPy [12]. Gröbner bases are computed effectively using computer algebra system Risa/Asir [14], connected to Python with OpenXM infrastructure for communicating mathematical software systems [10]. Then, the system of algebraic equations is solved using an appropriate solver called within Python. As the main focus of our paper, several solvers for solving a system of algebraic equations have been compared: an exact solver included in SymPy, a multivariate numerical solver using the secant method, and a univariate numerical solver with successive substitutions.

The rest of the paper is organized as follows. In Sect. 2, the method of inverse kinematics computation of a 3 DOF manipulator using Gröbner bases is explained. In Sect. 3, the description of the proposed system for solving the inverse kinematics problem is presented. In Sect. 4, the result of experiments is presented.

In this paper, an example of 3 degree-of-freedom (DOF) manipulator has been built using LEGO^® MINDSTORMS^® EV3 Education¹ (henceforth abbreviated to EV3) (Fig. 1). An EV3 set has a computer (which is called “EV3 Intelligent Brick”), servo motors and sensors (gyro, ultrasonic, color and touch sensors), along with bricks used for constructing building blocks of robots. While a GUI-based programming environment for controlling the robot is available, several programming languages such as Python, Ruby, C, and Java can also be used on the top of other programming environments.

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We first solve the forward kinematics problem. Components of the manipulator are defined as shown in Fig. 2. Segments (links) are called Segment i (\(i=1,2,3,4\)) from the one fixed on the ground towards the end effector, and a joint connecting Segment i and \(i+1\) is called Joint i. For Joint i, the coordinate system \(\mathrm {\Sigma }_i\), with the \(x_i\), \(y_i\) and \(z_i\) axes and the origin at Joint i, is defined according to the Denavit–Hartenberg convention [18] (Fig. 2). Note that, since the present coordinate system is right-handed, the positive axis pointing upwards and downwards is denoted by “\(\odot \)” and “\(\otimes \)”, respectively. Furthermore, let \(\mathrm {\Sigma }_0\) be the coordinate system satisfying that the origin is placed at the perpendicular foot from the origin of \(\mathrm {\Sigma }_1\), and the direction of axes \(x_0\), \(y_0\) and \(z_0\) are the same as that of axes \(x_1\), \(y_1\) and \(z_1\), respectively. Also, let Joint 5 be the end effector, and \(\mathrm {\Sigma }_5\) be the coordinate system with the origin placed at the position of Joint 5 and that the axes \(x_5\), \(y_5\) and \(z_5\) have the same direction as the axes \(x_4\), \(y_4\) and \(z_4\), respectively.

Let \(a_i\) be the distance between axes \(z_{i-1}\) and \(z_i\), \(\alpha _i\) the angle between axes \(z_{i-1}\) and \(z_i\) with respect to \(x_i\) axis, \(d_i\) the distance between axes \(x_{i-1}\) and \(x_i\), and \(\theta _i\) be the angle between axes \(x_{i-1}\) and \(x_i\) with respect to \(z_i\) axis. Then, the coordinate transformation matrix \({}^{i-1}T_i\) from the coordinate system \(\mathrm {\Sigma }_{i-1}\) to \(\mathrm {\Sigma }_i\) is expressed as

$$\begin{aligned} {}^{i-1} T_i&= \begin{pmatrix} 1 &{} 0 &{} 0 &{} a_i \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} \cos \alpha _i &{} - \sin \alpha _i &{} 0 \\ 0 &{} \sin \alpha _i &{} \cos \alpha _i &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} d_i \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \begin{pmatrix} \cos \theta _i &{} -\sin \theta _i &{} 0 &{} 0 \\ \sin \theta _i &{} \cos \theta _i &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} \\&= \begin{pmatrix} \cos \theta _i &{} -\sin \theta _i &{} 0 &{} a_i \\ \cos \alpha _i \sin \theta _i &{} \cos \alpha _i \cos \theta _i &{} -\sin \alpha _i &{} -d_i \sin \alpha _i \\ \sin \alpha _i \sin \theta _i &{} \sin \alpha _i \cos \theta _i &{} \cos \alpha _i &{} d_i \cos \alpha _i \\ 0 &{} 0 &{} 0 &{} 1 \end{pmatrix} . \end{aligned}$$

Since the joint parameters \(a_i\), \(\alpha _i\), \(d_i\) and \(\theta _i\) for EV3 are given as shown in Table 1 (note that the unit of \(a_i\) and \(d_i\) is centimeters) and the transformation matrix T from the coordinate system \(\mathrm {\Sigma }_5\) to \(\mathrm {\Sigma }_0\) is calculated as \(T={}^{0}T_1{}^{1}T_2{}^{2}T_3{}^{3}T_4{}^{4}T_5\), the position \({}^t(x,y,z)\) of the end effector with respect to the coordinate system \(\mathrm {\Sigma }_0\) is expressed as

$$\begin{aligned} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 8 (2 \cos (\theta _2 + \pi /4) + 2 \cos (\theta _2 + \theta _3 + \pi /4) + \sqrt{2}) \cos \theta _1 \\ 8 (2 \cos (\theta _2 + \pi /4 ) + 2 \cos (\theta _2 + \theta _3 + \pi /4)+ \sqrt{2} )\sin \theta _1 \\ 16 \sin (\theta _2 + \theta _3 + \pi /4 ) + 8 + 8 \sqrt{2} \end{pmatrix} . \end{aligned}$$

(1)

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Table 1.

Joint parameters for EV3.

i	\(a_i\) (cm)	\(\alpha _i\)	\(d_i\) (cm)	\(\theta _i\)
1	0	0	8	\(\theta _1 \)
2	0	\(\pi /2\)	0	\(\pi /4\)
3	16	0	0	\(\theta _2 \)
4	16	0	0	\(\theta _3\)
5	16	0	0	0

We have implemented a system for the inverse kinematics computation of the manipulator in SymPy [12] on the top of Python, connecting with the computer algebra system Risa/Asir [14] via OpenXM infrastructure for communicating mathematical software systems [10].

Python (and SymPy) has been chosen for rapid development including the use of the library for solving algebraic equations, and interoperability with the Robot Operating System (ROS) [9] for embedding our present implementation as a simulation environment or an inverse kinematics solver in the future.

OpenXM (which stands for “Open message eXchange protocol for Mathematics”) consists of definitions of protocols and data formats for communication and/or interchange of mathematical information among mathematical software systems. It also includes implementation of interface for various mathematical softwares including Risa/Asir, Kan/sm1 [20], Maple [11], Mathematica [26], MixedVol [3], NTL [16], PHC Pack [25] and TiGERS [4].

Risa/Asir is used for effective computation of Gröbner bases. After receiving input polynomials from Python/SymPy, it first computes the Gröbner basis with respect to the graded reverse lexicographic (grlex) ordering. Then, it converts the basis to the one with respect to lex ordering (with a modular FGLM algorithm [14]) before returning the final result to the host program. Risa/Asir can be called from Python easily using ctypes library [15] with the OpenXM interface library for Risa/Asir.

We have tested our implementation for inverse kinematics computation for randomly given points within the feasible region.

For each solver, 10 sets of experiments were conducted with 100 random points given in each set of experiments (thus 1000 random points were given in total).

The computing environment is as follows.

In this paper, we have presented a portable implementation of the inverse kinematics computation of a 3 DOF robot manipulator using Gröbner bases. The implementation is made on the top of Python and SymPy with using Risa/Asir for efficient computation of Gröbner bases and symbolic and/or numerical solvers called within Python for solving a system of algebraic equations. Risa/Asir can easily be called from Python via OpenXM infrastructure.

Thus, we see that univariate numerical solver with successive substitutions is effective for solving the inverse kinematics problem in the present case, although certification of real roots may be needed.

For future research, we need to improve implementation for calling Risa/Asir from Python via OpenXM in a more sophisticated way for more efficient computation.⁷ Furthermore, we intend to extend our implementation for embedding our solver in robotics simulators such as ROS and/or controlling the actual EV3 manipulators including the one we have built in the present paper.