# A Neural Network Based Approach to Inverse Kinematics Problem for General Six-Axis Robots
## 1 Introduction  
By virtue of recent breakthroughs in data communications, artificial intelligence and hardware, robots have experienced significant improvement in intelligent levels that prompt wide applications in daily lives, especially for repetitive, time-consuming, and high-risk tasks [1, 2, 3]. In such situations, the task is prescribed by a sequence of poses in the Cartesian space whereas the robot is controlled in the joint space. Hence, a problem termed the Inverse Kinematics Problem in robotics arose [4]. The IKP, together with its counterpart, the forward kinematic problem (FKP), gives a complete mapping between joint variables and the pose of the end-effector. Playing a key role in the robot trajectory planning and motion control, a solution for IKP with high precision is desirable. While FKP can be solved by the well known Denavit-Hartenberg (D-H) conventions with ease, handling IKP is much more challenging. Unlike FKP, different combinations of joint angles may lead to a same pose of the end-effector, resulting in multiple potential solutions. In fact, it has been proven that up to 16 inverse kinematics solutions can be expected for a general six degrees of freedom (DoFs) robot [5]. Furthermore, in order to reach any pose in the real space theoretically, robots are usually designed with at least six joints. Kinematic equations under such high DoFs are always highly coupled and non-linear, which poses substantial challenges for achieving the solution. Up to now, only robots with some special geometric structures, such as spherical wrist, are reported to have closed-form solutions [6, 7]. The IKP of this type can be decoupled and these robots are, hence, named decoupled robots. For others, e.g., the general typerobot, the closed-form solution does not exist. Numerous approaches based on different principles, which can be categorized as geometric [8; 9], algebraic [10; 11], and numerical methods [12; 13; 14], have been proposed. Unfortunately, all these methods suffer from inherent bottlenecks. Geometric methods are robot-dependent and only feasible for robots with simple structures. On the other side, algebraic methods fail to give an explicit expression of the final solution and the whole elimination processes are extremely tedious, making them hard and even impractical to be implemented under high-DoF circumstance. For numerical methods, apart from some possible drawbacks, such as singularity, joint limitations, and local optimum problem, the sought solution is deeply influenced by the initial guess set at the beginning. Strategies for choosing a suitable starting point add extra work. Due to the limitations of the above traditional approaches, soft-computing methods have gained more attentions and become increasingly popular in recent years. Via nonlinear function approximators or heuristic roots searching process [15], these algorithms can explore the answer without a priori knowledge of the robot models, among which neural network (NN) is a choice in fashion for IKP. Work of [PERSON] and [PERSON] on a 6 DoF robot was a typical early study [16]. Two architectures of artificial neural network (ANN)--one hidden layer with six outputs and two hidden layers with single output--were compared and the total errors were below 2% and 0.6%, respectively. However, the solutions were only represented within the joint space and not transformed into the Cartesian space, making the results less meaningful. Later, in [17], the performance of the radial basis function (RBF) network and the back propagation (BP) network for the IKP was compared. As the RBF network was concluded to outperform the BP network with a maximum error of 0.08\({}^{\circ}\), only three target poses were used during the test. Thus, room for improvement accompanies the result due to relatively low test data. To increase the reliability of the NN predictions, in [18], three recurrent networks were trained in parallel to form the prediction system.