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approaches have been investigated, not all of these approaches have been adopted in the physical telerehabilitation field, where the angles and trajectories of joints are usually utilised.

      In these two representations, the angle‐based approach is more widely used. This is mainly because human limbs are normally modelled as articulated rigid bodies. Additionally, some measurement devices, such as IMUs, are able to measure the orientation of limbs easily. Therefore, angles of joints and orientation of limbs can be acquired without much difficulty. Limb segments are hinged together with various degrees of freedom (DOF), which can be seen in Tables 1.1 and 1.2.

      (1.4)theta equals 180 minus cosine Superscript negative 1 Baseline left-parenthesis StartFraction c squared plus upper L squared minus a squared Over 2 c upper L EndFraction right-parenthesis minus cosine Superscript negative 1 Baseline left-parenthesis StartFraction d squared plus upper L squared minus b squared Over 2 d upper L EndFraction right-parenthesis comma

      (1.5)upper L squared equals a squared plus c squared minus 2 a c cosine alpha period

Schematic illustration of the locations of five sensors worn by a subject.

      Apart from the above three examples, in some studies where motion trajectories of joints are captured, angle information is still derived for encoding human movements. For example, Adams et al. [15] developed a virtual reality system to assess the motor function of upper extremities in daily living. To encode the movement, they used the swing angle of the shoulder joint along the Y and Z axes, the twist angle of the shoulder, the angle of the elbow, their first and second derivatives, the bone length of the collarbone, upper arm and forearm, as well as the pose (position, yaw and pitch) of the vector along the collarbone to describe the movement of the upper body. Here the collarbone is a virtual bone connecting two shoulders. These parameters were utilised in an unscented Kalman filter as state, while the positions of the shoulders, elbows and wrists reading from a Kinect formed the observation. Another example is that of Wenbing et al. [378], who evaluated the feasibility of using a single Kinect with a series of rules to assess the quality of movements in rehabilitation. Five movements, including hip abduction, bowling, sit to stand, can turn and toe touch, were studied in this paper. For the first four movements, angles were used as encoders. For instance, the change of angle between left and right thighs (the vector from the hip centre to the left and right knee) was used to represent the angle of hip abduction, while the dot product of two vectors (from the hip centre to the left and right shoulders) was utilised to compute the angle encoding the movement of bowling. Additionally, Olesh et al. [267] proposed an automated approach to assess the impairment of upper limb movements caused by stroke. To encode the movement of the upper extremities, the angle of four joints, including shoulder flexion‐extension, shoulder abduction‐adduction, elbow flexion‐extension and wrist flexion‐extension, were calculated with the 3D positions of joints measured with Kinect.

      Though angles of joints, as well as their derivatives, are utilised widely in encoding human motions, trajectories of joints and their derivatives can also be observed in some rehabilitation and telerehabilitation applications.

      1.5.3 Summary and challenge

Photo depicts the pictures of animals.

      As a result, there remain challenges in developing formal descriptions and robust computational procedures for the automatic interpretation and representation of motions of patients. The majority of studies [92, 158] employed a variety of human motion encoders to recognise or decompose general movement, such as reaching, waving hands, jumping, walking and so on. Few of them investigated details in each general movement, for example, the even smaller atomic components included in these general movements that are of importance for syntactic and structural descriptions of human movements in detail, especially in a clinic and rehabilitation environment, where the details of movements of body parts require a form of motion language or, at least, syntax. A novel approach to encode human motion trajectories will be discussed in Chapter 3.

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