# A Wheeled Inverted Pendulum Learning Stable and Accurate Control from Demonstrations
## 2 Related Work  
By conventional control methodologies, the robot task is usually encoded by manually designing a controller [14; 15; 16]. In order to acquire more complicated control strategy from humans automatically, learning approaches are applied to yield a controller in the data-driven fashion. Spline decomposition methods [17; 18] are presented to construct the robot regulators by computing the point-wise averages of the demonstrated trajectories. These methods are efficient and simple to execute, but they depend mainly on the human heuristics for trajectory synthesis. Statistical approaches [19; 20] are also prevailing for modeling robot motions. These approaches usually need a reasonable heuristic for controller programming, therefore, being possibly influenced by external perturbations. These approaches are close but still unable to generalize to similar task cases in that the choice of heuristic is decided based on specific task parameters. The dynamic movement primitives (DMP) were proposed in [21] to effectively address the instability problem. Its main idea is to approximate a nonlinear system by several linear counterparts. In contrast to its effect on the aspect of stabilization, DMP did not emphasize on the accuracy factor when modeling complicated tasks or handling complex scenarios. Dynamic systems (DS) based methods [22; 23] were developed as a promising alternative to traditional control approaches due to its convenience, which can extract human control strategies accurately and automatically from given demonstrations. Several methods choose to model dynamic systems without considering stability constraints, which makes them possibly unstable at a target position [24; 25; 26]. A stable estimator of a dynamic system (SEDS) was proposed in [27] with Gaussian mixture models (GMMs) to incorporate both accuracy and stability factors. SEDS uses the Lyapunov stability condition as a constraint to optimize the parameters in the control model. It effectively captures the dynamics features from human demonstrations whilst stabilizing the robot motions. Fast and stable learning of dynamic system method (FSM-DM) in [28] handles three factors for learning a dynamic system, which includes one more factor of learning speed in addition to accuracy and stability. FSM-DM is able to teach a system fast, which is advantageous in practice when efficiency is a important requirement. In order to reduce the influence of the accuracy against the stability dilemma, the algorithm of control-Lyapunov-function-based dynamic movements (CLF-DM) was proposed in [29]. This method is divided into three steps. First, A parameterized Lyapunov function candidate is roughly taught to be consistent with the data. Second, the control features reflected by the demonstrations are learned to guarantee the accuracy factor. Third, an online correction technique is developed to stabilize the reproduced trajectory on the fly. The neurally imprinted stable vector fields (NIVF) technique was presented as a learning framework to incorporate stable vector fields into the extreme learning machine (ELM) [30], which reproduces more accurate motions and is finally stable at the target position. The approach proposed in [31] considered using ELM network to train a [PERSON] candidate based on a specific task. The learned Lyapunov candidate is consistent with the motion trajectories related to the task; therefore, making the stability constraint based on it more slack. Accordingly, the accuracy of the learned model is also less influenced by the stability factor. \(\tau\)-SEDS was presented in [32] to handle the trade-off of stability and accuracy by means of diffeomorphism. It transforms the original robot motions into those in a new space, where the transformed motions are consistent with a quadratic Lyapunov candidate. The method (MIMS) in [33] presented extended the SEDS scheme under the manifold immersion and submersion.