SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 29: Observation space of Hand Over.
Index	Description
0 - 373	dual hands observation shown in Table 28
374 - 380	object pose
381 - 383	object linear velocity
384 - 386	object angle velocity
387 - 393	goal pose
394 - 397	goal rot - object rot
Actions The action space for a single hand in the Hand Over task comprises 40 dimensions, as illustrated in Table 30.
Table 30: Action space of Hand Over.
Index	Description
0 - 19	right Shadow Hand actuated joint
20 - 39	left Shadow Hand actuated joint
Rewards Let the positions of the object and the goal be denoted as  $x_{o}$  and  $x_{g}$  respectively. The translational position difference between the object and the goal, represented as  $d_{t}$ , can be computed as  $d_{t} = \| x_{o} - x_{g}\|_{2}$ . Similarly, we define the angular position difference between the object and the goal as  $d_{a}$ . The rotational difference, denoted as  $d_{r}$ , is then calculated as  $d_{r} = 2\arcsin(\mathrm{clamp}(\|d_{a}\|_{2}, \max = 1.0))$ .
The rewards for the Hand Over task are determined using the following formula:
$$
r = \exp (- 0. 2 (\alpha d _ {t} + d _ {r})) \tag {2}
$$
Here,  $\alpha$  represents a constant that balances the rewards between translational and rotational aspects.
Hand Over Catch
This environment is made up of a half Hand Over, and Catch Underarm [42], the object needs to be thrown from the vertical hand to the palm-up hand.
Observations The observational space for this combined task encompasses 422 dimensions, as illustrated in Table 31.
Table 31: Observation space of Hand Over Catch.
Index	Description
0 - 397	dual hands observation shown in Table 28
398 - 404	object pose
405 - 407	object linear velocity
408 - 410	object angle velocity
411 - 417	goal pose
418 - 421	goal rot - object rot
Actions The action space, consisting of 52 dimensions, is illustrated in Table 32, providing a comprehensive representation of the available actions.
Table 32: Action space of Hand Over Catch.
Index	Description
0 - 19	right Shadow Hand actuated joint
20 - 22	right Shadow Hand base translation
23 - 25	right Shadow Hand base rotation
26 - 45	left Shadow Hand actuated joint
46 - 48	left Shadow Hand base translation
49 - 51	left Shadow Hand base rotation
Rewards Let's denote the positions of the object and the goal as  $x_{o}$  and  $x_{g}$ , respectively. The translational position difference between the object and the goal denoted as  $d_{t}$ , can be calculated as  $d_{t} = \| x_{o} - x_{g}\|_{2}$ . Additionally, we define the angular position difference between the object and the goal as  $d_{a}$ . The rotational difference, denoted as  $d_{r}$ , is given by the formula  $d_{r} = 2\arcsin(\mathrm{clamp}(\|d_{a}\|_{2}, \max = 1.0))$ . Finally, the rewards are determined using the specific formula:
$$
r = \exp \left[ - 0. 2 \left(\alpha d _ {t} + d _ {r}\right) \right] \tag {3}
$$
Here,  $\alpha$  represents a constant that balances the translational and rotational rewards.
C.3 Constraint Specification
(d): Safety Joint
Figure 23: Tasks of Safety-DexterousHands.
Safety Joint constrains the freedom of joint 4 of the forefinger (please refer to Figure 23 (c) and (d)). Without the constraint, joint 4 has freedom of  $[-20^{\circ}, 20^{\circ}]$ . The safety tasks restrict joint 4 within  $[-10^{\circ}, 10^{\circ}]$ . Let ang_4 be the angle of joint 4, and the cost is defined as:
$$
c _ {t} = \mathbb {I} (\text {a n g} - 4 \notin [ - 1 0 ^ {\circ}, 1 0 ^ {\circ} ]). \tag {4}
$$
Safety Finger constrains the freedom of joints ②, ③ and ④ of forefinger (please refer to Figure 23 (c) and (e)). Without the constraint, joints ② and ③ have freedom of  $[0^{\circ}, 90^{\circ}]$  and joint ④ of  $[-20^{\circ}, 20^{\circ}]$ . The safety tasks restrict joints ②, ③, and ④ within  $[22.5^{\circ}, 67.5^{\circ}]$ ,  $[22.5^{\circ}, 67.5^{\circ}]$ , and  $[-10^{\circ}, 10^{\circ}]$  respectively. Let ang_2, ang_3, ang_4 be the angles of joints ②, ③, ④, and the cost is defined as:
$$
c _ {t} = \mathbb {I} (\text {a n g} _ {- 2} \notin [ 2 2. 5 ^ {\circ}, 6 7. 5 ^ {\circ} ], \text {o r a n g} _ {- 3} \notin [ 2 2. 5 ^ {\circ}, 6 7. 5 ^ {\circ} ], \text {o r a n g} _ {- 4} \notin [ - 1 0 ^ {\circ}, 1 0 ^ {\circ} ]). \tag {5}
$$