Attributes of Wielded Objects

Human Hand Function 94 page

좀 흥미로워서 따로 정리해봄.

Turvey and his colleagues have presented a substantial body of empirical evidence to support their argument that “the spatial capabilities of dynamic touch result from the sensitivity of the body's tissues to certain quantities of rotational dynamics about a fixed point that do not vary with changes in the rotational forces (torques) and motions”(1996,p.1134)

(중략)

One final issue that has been frequently raised by opponents of dynamic touch and its theoretical underpinnings pertains to the absence of any attempt to relate the critical mathematical constructs in their theory to neurophysiology. There are no known biological sensors in muscles that respond to the inertia tensor or any of its components, whether singly or in some combination. How then does the observer/actor directly determine eigen vectors and eigen values as required by the theory?

다 읽어보면, 피부조직의 어떤 신경 세포가 dynamic touch (wielding) 과정에서 rotationial inertia를 센싱을 해서 물체의 길이 같은 attribute를 사람이 알게 된다는 이론이다. 근데 그게 어떤 신경 세포인 지 실험이나 이론이 없어서 근거가 부족함. 

Keywords:

• Dynamic Touch: A haptic subsystem involving the perception of objects through kinesthetic sensing when an object is wielded (moved, rotated, etc.).
• Inertia Tensor: A 3x3 matrix representing resistance to rotational acceleration around different axes.
• Eigenvalues and Eigenvectors: Mathematical components of the inertia tensor that correspond to the principal moments of inertia and the primary axes of the object.
• Perceptual Judgment: How people assess attributes like length, weight, and shape based on the resistance to rotation.

Takeaways:

• Dynamic Touch & Perception: The study explores how individuals perceive the physical properties of objects through their resistance to rotational movement, not just mass or volume.
• Inertia Tensor Role: The inertia tensor’s eigenvalues and eigenvectors help map perceived attributes like length or weight to physical properties of objects.
• Haptic Size-Weight Illusion: The perception of heaviness is influenced by rotational inertia rather than cognitive inference, suggesting a direct link between rotational resistance and perceived weight.
• Kinesthetic Sensing: This form of touch involves sensing muscle, tendon, and ligament deformation during manipulation, playing a fundamental role in motor control beyond traditional touch.
• Application in Tool Use: Insights from dynamic touch could be valuable for understanding how humans use tools, particularly in tasks involving teleoperation.

The hypothesis that perceived length is a function of rotational inertia can be tested by altering the mass distribution of rods that are of the same length.

Understanding Eigenvalues and Eigenvectors in the Context of Rotational Inertia

**1. Basics of Eigenvalues and Eigenvectors:

• Eigenvalues and eigenvectors are mathematical concepts from linear algebra, often used to simplify complex systems, like the inertia tensor in rotational dynamics.
• Eigenvectors: These are vectors that define specific directions in a system where the associated linear transformation (like rotation) acts only by stretching or compressing, not changing the direction of the vector.
• Eigenvalues: These are the factors by which the eigenvectors are stretched or compressed. In the context of rotational inertia, they represent the resistance to rotational acceleration along the eigenvector's direction.

**2. Inertia Tensor and Rotational Dynamics:

• The inertia tensor is a 3x3 matrix that quantifies how an object resists rotation around different axes.
• This matrix can be complex because it considers how mass is distributed relative to different rotational axes.
• Principal moments of inertia are the eigenvalues of this tensor. They indicate the resistance to rotation around specific, principal axes (which are the eigenvectors).

**3. Example 1: Simple Rod

• Imagine a uniform rod.
• Axes: Consider three axes for rotation: along the length of the rod (axis 1), and perpendicular to the rod at its midpoint (axes 2 and 3).
• Eigenvectors: The primary axes (1, 2, 3) are the eigenvectors because they represent the directions in which the rod's rotation can be analyzed independently.
• Eigenvalues: For rotation around axis 1 (along the length), the eigenvalue is small because the rod offers little resistance (low moment of inertia). For rotation around axes 2 and 3 (perpendicular to the rod), the eigenvalues are larger because the mass distribution is farther from these axes, offering more resistance (higher moments of inertia).

**4. Example 2: Dumbbell

• Consider a dumbbell with two equal masses connected by a rod.
• Axes: One axis passes through the rod (axis 1), and the other two are perpendicular to the rod (axes 2 and 3).
• Eigenvectors: The rod's length and the perpendicular directions are eigenvectors because rotation around these axes represents distinct rotational behaviors.
• Eigenvalues: Rotation around axis 1 (through the rod) has a small eigenvalue (low moment of inertia) since the masses are close to this axis. For axes 2 and 3 (perpendicular), the eigenvalues are large (high moments of inertia) because the masses are farther from these axes, making rotation harder.

Key Takeaways:

• Eigenvectors: Represent the principal axes around which an object’s rotation can be analyzed independently.
• Eigenvalues: Represent the amount of resistance to rotational acceleration around these axes.