# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 1 Introduction  
Harmonic drives (HDs), which were invented in the 1950s [1], are well known for their high gear-reduction ratio, low weight, small volume, lack of backlash, and high efficiency [2]. Therefore, HDs are used widely in intelligent robotics and various high-tech fields [3]. Many studies have been conducted on HD kinematic error [4], nonlinear stiffness [5; 6; 7; 8; 9; 10], hysteresis [11; 12; 13; 14; 15; 16], and dynamic friction [17]. However, no definite understanding of the mechanism of transmission compliance and hysteresis behaviors or their degradation is available in the literature. In previous works, the HD is considered to be a black box or grey box, which causes the accuracy of the HD model to degrade over time. As stated in a manufacturer's catalog [5], the torque-displacement relationship for an HD consists of two characteristic properties: (i) the increasing stiffness with torsional angle, and (ii) the hysteresis characteristic. To capture the nonlinear stiffness, the manufacturer suggests using a piecewise linear function of the torsion angle [5]. Some other scholars prefer to approximate the stiffness curve with a cubic or quintic polynomial [6; 7; 8]. Even in certain applications, linear stiffness can be sufficient to simulate the behavior of the reducer [9]. Recently, [PERSON] et al. [10] proposed the HD rotational compliance model by modeling the compliance behaviors of the flexspline and the wave generator. Considering an HD mechanical system analogy, the transmission compliance is described by the behavior of a serial spring-mass system. Hysteresis is widespread in the fields of smart materials, magnetic fields, and mechanical systems. The hysteresis in HDs is considered to be caused by the combined effect of torsional flexibility and friction [8; 11]. [PERSON] et al. [12] proposed a hysteresis model, represented by a hyperbolic function, in which the hysteresis loss is estimated as a combination of Coulomb friction and a weighted friction function. Assuming thatthe hysteresis mainly arose from structural damping, [PERSON] [13] developed a model to describe the hysteresis of an HD. Based on the heredity concept of dynamic systems, a mathematically well-posed nonlinear differential equation was proposed by [PERSON] et al. [8] to describe the hysteresis phenomenon in HDs. [PERSON] et al. [14] used the Preisach model to capture the hysteretic behavior of HDs. [PERSON] et al. [15] proposed a new type of comprehensive harmonic drive model. They used the Maxwell resistive-capacitor model to model the hysteresis behavior in harmonic drives that are taken to be black boxes. [PERSON] et al. [16] developed a grey-box model for HDs that captured the hysteresis behavior by combining nonlinear stiffness characteristics and distributed Maxwell-slip elements. [PERSON] et al. [10] recently proposed a simple hysteresis model. A hysteresis model considering sheath stiffness was proposed by [PERSON] et al. [18] to capture the hysteretic behavior of the tendon sheath mechanism. Interferences occur at the contact teeth even at no load in HDs [19]. [PERSON] et al. [20] proposed a method of estimating the load shared by the multiple tooth pairs in contact. They pointed out that the interference almost vanished with the increase of the external load. The number of effective meshing teeth pairs increases sharply with the increasing resistance torque [21]. Many experiments also have been done on the lubrication failure of HDs [22; 23; 24; 25]. These experimental results showed that obvious wear was found on the wave generator-flexspline interface and the meshing tooth surface. However, these mechanical characteristics of HDs were not considered in the previous grey- or black-box models. This could cause the applicability of the HD model to deteriorate with the degradation of the reducer. The objective of this study was to develop a mechanical model of transmission compliance and hysteresis in HDs for high-performance robot joint control.