# Mapping of the Upper Limb Work-Space: Benchmarking Four Wrist Smoothness Metrics
## 1 Introduction  
Smoothness usually describes the quality of a movement and is recognized as a feature of coordinated and skilled motor behavior [1], characterizing goal-directed movements [2]. Movement smoothness is commonly used as a measure of motor performance and motion control with wide application in healthy subjects [3] and neurological patients [4, 5]. Physiological motion is characterized by high smoothness in the upper limb workspace. It was proposed that smoothness is a motion planning criterium for determining limb trajectories [6, 7]. A reduced movement smoothness reflects deficits in movement planning and execution. Therefore, smoothness has become an important metric for the evaluation of patients with deficits in motor control and for the assessment of their recovery [8], such as patients suffering from stroke [9], cerebral palsy [10], Parkinson's disease [11], or multiple sclerosis [12].  
The most frequently used methods for evaluating movement performance in clinical scenarios are clinical scales and functional tests, e.g., Jebsen-Taylor Hand Function (JTHF) [13], Arm Research Assessment Test (ARAT) [14], and Nine-Hole Peg test (NHPT) [15]. Although these measures are easy to administer, they are partially dependent on the observers that score the tests [16]. Therefore, objective smoothness metrics based on kinematics have beendeveloped. Smoothness can be measured with indexes referred to wrist motion. These indexes are based on the velocity profile of the wrist and its derivatives and include: the number of velocity peaks [17,18], the speed metric [19,20], defined as the ratio between the mean and the maximum velocity; the jerk, which is the third order derivative of the position profile, and related indexes such as the normalized jerk [11], the log dimensionless jerk [21], or the normalized squared jerk [22]; the arc length of the velocity or of the power spectrum [21,23].  
Jerk is also frequently used as a measure for motion dexterity [24]. Kinematic assessment can detect changes in movement performance that cannot be captured by clinical scales and tests and provides a more objective measure [25]. Kinematics measures have been compared to clinical scales usually adopted for the evaluation of patient performance, proving their validity as indicators of smoothness and motion dexterity [25,26]. Therefore, smoothness analyses may help clinicians in the evaluation of dexterity and motor control.  
However, so far, smoothness has not been investigated in a comprehensive set of directions in the workspace of the upper limb, and comparative maps of smoothness based on multiple indexes have not been created before. In this paper, we aim at (i) reconstructing a map of smoothness in the upper limb workspace of healthy subjects that can be used as reference data in clinical applications and (ii) comparing several metrics based on wrist motion. First, fifteen subjects performed a variety of point-to-point reaching movements in multiple directions in five planes of motion. Then, the normalized jerk (NJ), the speed metric (SM), the spectral arc length (SAL), and the number of speed peaks (NP) were computed in order to evaluate the smoothness in each movement direction. A smoothness map was reconstructed in both the 3D space and with respect to shoulder articular angles and the difference between sectors were evaluated. Finally, the correlations between pairs of metrics were computed.