Jayanta Kumar Dutt

Dynamics of Tree-Type Robotic Systems

As reviewed in Chap. 2, Newton-Euler (NE) equations of motion are found to be popular in dynamic formulations. Several methods were also proposed by various researchers to obtain the Euler-Langrage’s form of NE equations of motion. One of these methods is based on velocity transformation of the kinematic constraints, e.g., the Natural Orthogonal Complement (NOC) or the Decoupled NOC (DeNOC), as obtained in Chap. 4. The DeNOC matrices of Eq. (4.28) are used in this chapter to obtain the minimal order dynamic equations of motion that have several benefits.

Euler-Angle-Joints (EAJs)

As frequently noted in the literature on robotics (Sugihara et al. 2002; Kurazume et al. 2003; Vukobratovic et al. 2007; Kwon and Park 2009) and mechanisms (Duffy 1978; Chaudhary and Saha 2007), a higher Degrees-of-Freedom (DOF) joint, say, a universal, a cylindrical or a spherical joint, can be represented using a combination of several intersecting 1-DOF joints. For example, a universal joint also known as Hooke’s joint is a combination of two revolute joints, the axes of which intersect at a point, whereas a cylindrical joint is a combination of a revolute joint and a prismatic joint. Similarly, the kinematic behavior of a spherical joint may be simulated by the combination of three revolute joints whose axes intersect at a point. The joint axes can be represented using the popular Denavit and Hartenberg (DH) parameters (Denavit and Hartenberg 1955). For the spherical joints, an alternative approach using the Euler angles can also be adopted, as there are three variables. For spatial rotations, one may also use other minimal set representations like Bryant (or Cardan) angles, Rodriguez parameters, etc. or non-minimal set representation like Euler parameters, quaternion, etc. The non-minimal sets are not considered here due the fact that the dynamic models obtained in this book are desired in minimal sets. The minimal sets, other than Euler/Bryant angles, are discarded here as they do not have direct correlation with the axis-wise rotations. It is worth mentioning that the fundamental difference between the Euler and Bryant angles lies in a fact that the former represents a sequence of rotations about the same axis separated with a rotation about a different axis, denoted as α–β–α, whereas the latter represents the sequence of rotations about three different axes, denoted as α–β–γ. They are also commonly referred to as symmetric and asymmetric sets of Euler angles in the literature. For convenience, the name Euler angles will be referred to both Euler and Bryant angles, hereafter.

Transient Excitation Suppression Capabilities of Electromagnetic Actuators in Rotor-Shaft Systems

Impulsive forces cause sudden variations in dynamic systems and may result in fatal damage to the system especially in turbines where the whole system is enclosed inside a casing with very small clearance between the blades and the casing. Electromagnetic actuators have been known to apply non-contact electromagnetic force on a rotor- shaft section preventing rotor-shaft vibrations. This work attempts to investigate the comparison of two different control laws in controlling the vibrations caused by noises such as sudden flow rate change, blade loss, or excitations occurring due to seismic vibrations. A rotor-shaft system is developed within a simulation framework which includes an actuator placed away from the bearings. Literature shows the use of conventional PD (Proportional Derivative) control law which is equivalent to a 2-element support model. This work novels the 3-element viscoelastic support model, which is found to offer better vibration mitigation abilities in terms of controlling transient excitations. Preliminary theoretical simulation using linearized expression of electromagnetic force and the accompanying example show good reduction in transverse response amplitude, postponement of instability caused by viscous form of rotor internal damping.

Vibration Control of Rotor Shaft Systems Using Electromagnetic Actuator

Applicability of electro-magnetic actuator for active vibration control of long rotors, commonly found in power-plant turbines is addressed in this work through simulations. Appreciable vibration reduction with the Proportional-Derivative (PD) control law applied on the difference between nominal and instantaneous air-gap between rotor surface and actuator poles is reported in literature. This paper investigates and finds that the Proportional and high frequency band limited Derivative control action (abbreviated as PDhfbl) is much more efficient than the former in terms of unbalanced response reduction and increment of Stability Limit Speeds (SLS) of long rotor-shaft systems. This paper also establishes a philosophical link between the PDhfbl control action and a multi-element mechanical suspension model, or viscoelastic suspension model, reported to reduce response and increase stability limit speed of rotors in literature. Examples of simulations are presented in support of the conclusions.

Stability Study for Vibration Control of an Active Magnetic Bearing Supported Rotor Mounted on a Moving Base

Active Magnetic Bearings (AMBs) suspend rotor systems and also facilitate active rotor vibration control through feedback control law for appropriate contactless electromagnetic force. However for rotors on moving bases e.g. ships, airplanes, the equations of rotor motion become parametrically excited and so a feedback control methodology, e.g. with Proportional Derivative (PD) control law needs assessment of stability. It is observed that the equations of motion of a rigid rotor with respect to base, translating and pitching in one plane, under the action of linearized electromagnetic forces from AMBs at ends, take the form of forced damped Mathieu equations. This paper first attempts the stability analysis to tune controller parameters and then uses the Active Disturbance Rejection Control (ADRC) with Extended State Observers (ESO) to reject disturbances. Simulated results show considerable rotor vibration suppression and thus prove AMBs as effective suspensions for vibration control of rotor-shaft systems on moving platforms.

Multi Speed Model Updating of Rotor Systems

Accurate Finite Element (FE) models of rotor systems are required for predicting its dynamic behavior, in dynamic design and fault identification purposes. In inverse eigen-sensitivity method of finite element model updating, the limited number of measured eigenvalues available at any spin speed restricts the maximum number of parameters that can be updated. This paper proposes a multi speed model updating method based on inverse eigenvalue sensitivities to update parameters of a rotor system. The method uses eigenvalues obtained at more than one spin speed to update the model. Such an approach allows not only to update more number of parameters but also helps in obtaining a more consistent estimate of updating parameters.

Dynamics of Robotic Systems

The field of robotics has grown a lot in last three to four decades. In this chapter, background and developments in the field of dynamics of robotic systems are presented.

Recursive Dynamics for Floating-Base Systems

Robotic systems studied in Chap. 6 have their bases fixed, however, in reality many robotic systems have their bases mobile or floating. In the case of a fixed-base robotic system, the base does not influence the dynamics, whereas it significantly influences the dynamics in the case of a floating-base robotic system. Space manipulators and legged robots are examples of floating-base robotic systems. Legged robots find applications in maintenance task of industrial plants, operations in dangerous and emergency environments, surveillance, maneuvering unknown terrains, human care, terrain adaptive vehicles and many more. In the case of legged robots they are either classified based on the number of legs, e.g., biped, quadruped, hexapod, etc., or the way it balances, e.g., statically or dynamically balanced. As reviewed in Chap. 2, legged robots (1) have variable topology, (2) move with high joint accelerations, (3) are dynamically not balanced if Center-of-Mass (COM) moves out of the polygon formed by the support feet, and (4) are under actuated. Hence, objective of achieving stable motion is difficult to decompose into actuator commands. Therefore, control of legged robots is intricate and dynamics plays vital role in achieving stable motion.

Kinematics of Tree-Type Robotic Systems

Kinematic modeling of a tree-type robotic system is presented in this chapter. In order to obtain kinematic constraints, a tree-type topology is first divided into a set of modules. The kinematic constraints are then obtained between these modules by introducing the concepts of module-twist, module-joint-rate, etc. This helps in obtaining the generic form of the Decoupled Natural Orthogonal Complement (DeNOC) matrices for a tree-type system with the help of module-to-module velocity transformations. Using the present derivation, link-to-link velocity transformation (Saha 1999a, b) turns out to be a special case of the module-to-module velocity transformation (Shah et al. 2012a) presented in this chapter.

Recursive Dynamics for Fixed-Base Robotic Systems

In this chapter, dynamic analyses of fixed-base robotic systems are presented using the dynamic modeling presented in Chap. 5. For this, recursive inverse and forward dynamics algorithms are developed. The algorithms take care of the multiple-DOF joints in an efficient manner, as explained in Sect. 4.2.1; in contrast to treating them as a combination of several 1-DOF joints by taking into account the total number of links equal to number of 1-DOF joints or joint variables. In the presence of many multiple-DOF joints in a robotic system the latter approach is relatively inefficient due to the burden of unnecessary computations with zeros. The improvement in the computational efficiency in the presence of multiple-DOF joints are addressed in this chapter. Dynamic analyses, namely, the inverse and forward dynamics, of several systems are performed in this chapter.