# Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters
## 2 Relevant Work  
DMPs are a robotic control framework that uses a series of underlying kernels--attractor points or limit cycles--to produce trajectories [10; 11; 12; 34]. The strength, timing, and growth/decay of the attractors can be varied to create custom trajectories [35]. Over the years, DMPs have been adjusted to better learn rhythmic movements [35; 36], to learn from multiple non-identical demonstrations [37], and to learn joint torques along with the kinematic trajectory [38], among other expansions [11; 39; 40]. In an effort to improve the versatility of DMPs in online applications, [PERSON] et al. introduced the DMP+ framework in 2016 that can partially update the DMP+ kernel weights to locally adjust a learned trajectory [10]. To update DMP+ kernels, the weights must be relearned according to the user's chosen algorithm, whereas SMP kernel updates can be achieved using the SMP visualization feature [1] and/or the system parameters described in this article. Both SMP update methods maintain the computational complexity of an update step, while the DMP+ method may increase in complexity depending on the desired task.  
SHCs are a series of saddle equilibria where the unstable manifold of one equilibrium point leads onto the stable manifold of another; this creates pathways between the saddle points (see Figure 1) [7; 13; 14]. SHCs have been used as a model for neural activation patterns in animals [23; 41; 42]. They have also been used to produce and investigate dynamical state systems [5; 43], and apply those systems to robotic movement [14; 28; 44]. In 2015, [PERSON] et al. described the system parameters alpha \(\alpha\), beta \(\beta\), and nu \(\
u\) for SHCs [7]. In their work, they described alpha as the growth rate of a kernel--how fast the kernel grows in its respective dimension. Beta was described as the kernel magnitude--the maximum amplitude of the waveform. Nu was described as the saddle value--a kernel's insensitivity to external perturbation (modeled as noise). In the SMP system, these variables (and noise) are varied synchronously to create kernels that remain connected in state space, i.e., the connected kernels create a smooth trajectory in the task space. SMPs expand SHCs into a stable, learnable system with a clear transformation from state space into a robot's task space.  
Mathematical representations of biological systems are commonly used to develop robot control frameworks [8; 18; 24; 40; 45; 46]. These frameworks have value in both biological and engineering applications, and characterizing their parameters increases ease-of-use in either application [47; 48; 49]. Some biologically inspired, transient dynamic systems, like neural networks, have gained much popularity in recent years, but at larger scales, system analysis, parametrization, and explainability become nearly impossible [26; 39; 50; 51]. Like these other frameworks, SMPs have a biological relevance because of their construction from biologically relevant SHCs. Unlike these frameworks, SMPs are parameterizable because their parameters--from DMPs and from SHCs--have already been described separately in each framework [7; 34].  
Figure 1: Stable heteroclinic channel (SHC) cycle with 3 saddle node points. The cycle is formed by connecting the unstable eigenvector of a saddle onto the stable eigenvector of another saddle. The Lotka–Volterra formulation of SHCs requires at least 3 saddles in the cycle.