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y0
z0=xr×y0
kxr×y0k
x0=y0×z0
x0z0
xr
O=OrOr
(O, x0,y0)
xr=Rr0(z0, φr)x0
Rr0(z0, φr)φr
z0
Rab(v, θ)Rab
b a v
θ
yr=Rr0(z0, φr)y0
zr=Rr0(z0, φr)z0=z0
xh
yh=yh,pxh
xh,p
xh
yh,p=Rhr,1(z0, φh(φr)) yr
xh,p=Rhr,1(z0, φh(φr)) xr
yh=yh,p
xh=Rhr,2(yh,p, ψh(φr)) xh,p
xhxr
φr
xh(O, x0,y0)
φh(φr)
Rhr,2(yh,pψh(φr))
Wh
h=Wh,p
h=Rh0(z0, φr+φh(φr))TW0
h
Wh
hW0
h
φrφh(φr)
zr
(Oh,p,xh,p)
xh
(O, x0,y0)
xr
yr
φr
φr=q
q
τhq
τh= (xgxh×Wh)·z0
=−mf,h gxgcos(q+φh(q))
τh
(Oh,xh)mf,h
xg
q
φh(q)
F=mf,h
xg
li
gcos(q+φh(q))
cos(φh(q))
F
li
xh
(Oh,xh)
Or
Or
τi=lF
=lmf,h
xg
li
gcos(q+φh(q))
cos(φh(q))
τi
l
Or
φhφh,0
τi,wjm =lmf,h
xg
li
gcos(q)
τi,wjm
φh(q)
φh(q)≈φh
φh+q > 40◦φh+q < −60◦
φh(q)
ˆmeq,h
ˆ
φh= arg min
meq,h,φhF−meq ,hgcos(q+φh)
cos(φh)2
ˆmeq,h ˆ
φhmeq,h
φhmeq,h =mf,h xg/li
F q
17
7 10
70.7±9.7kg 176 ±7.8cm
24 ±4.9
25.5±1.4cm 18.6±1.3cm
2017 34
20
[−85.94◦,39.53◦]
5s
1kHz
φ,0
MAE =1
NX|ˆ
F(q, φ)−F|
Nˆ
F(q, φ)
F
t
p < 0.05
τi=αlmeq,h gcos(q+φh)
cos(φh)
α
α
1 0
τctrl =Kp(ˆτi(q, φ)−τi)
+KiZt
0
(ˆτi(q, φ)−τi)dt + ˆτcomp (¨q, ˙q, q)
τctrl
ˆτi(q, φ)
τi
ˆτcomp (¨q, ˙q, q)
¨q˙q q
ˆτcomp (¨q, ˙q, q) = ˆ
M(q)¨q+ˆ
C(q, ˙q)+ ˆ
G(q)+sign( ˙q)ˆτC+ˆν˙q
ˆ
M(q)
ˆ
C(q, ˙q)
ˆ
G(q)
ˆτC
ˆν
ˆτcomp
τF F,τ =KF F,τ ˆτi(q, φ) + τF F ,τ 0
τF F,τ
KF F,τ
ˆτi(q, φ)
τF F,τ 0
l
τF F,q =KFF,q q+τF F,q 0
τF F,q
KF F,q
q
τF F,q 0
τctrl =Kp(ˆτi(q, φ)−τi)
+KiZt
0
(ˆτi(q , φ)−τi)dt + ˆτcomp ( ˙q, q)
+τF F,τ +τFF,q
α
60◦
F
N= 8
kg cm cm cm
23.2±3.0 67.1±11.8 174.7±7.6 25.5±1.6 18.8±1.0
24 ±4.9 70.7±9.7 176 ±7.8 25.5±1.4 18.6±1.3
24.3±2.2 70.7±9.8 176 ±8.4 26.4±2.0 19.5±1.2
N
meq,h (kg)φh(rad)
2.13 ±0.56 0.27 ±0.12
1.93 ±0.26 0.21 ±0.07
1.90 ±0.36 0.23 ±0.07
1010
7kHz
Tup Tbot Tmid
60◦
1s
600 ms
α= 1
2s
7 30 15
15 210
6
6
4 30
15 15
120
Tup
Tbot Tmid
3
3
τi
τi=lF
l
F
l
F τi
α= 1
τi>0Nm
t−
p < 0.05
N N
φ,0 1.9 0.68
φ <0.62 0.28
meq,h ≈2.45 kg ;φ≈0.217 rad
meq,h ≈2.14 kg
t
T−7.18
p= 3.91 ×10−7
3
2.4
meq,h φh
meq,h kg φhrad
1.93 0.21
0.26 0.07
φh
Tup
Tbot Tmid
Tmid Tup
F2,36 = 36.7p= 0.3×10−5η2= 0.67
p= 1.1×10−4p= 0.7×10−5
p= 2.5×10−3
F
τi
27%
24%
44%
p= 2.49 ×10−11, F =
54.5t−
p= 9.3×
10−5, T = 5.61 p= 5.54 ×
10−8, T = 9.88 p=
2.85 ×10−4, T = 5.07
F3,54 = 0.446 p= 0.721
η2= 0.0242
rad/s rad/s2s
29.1±6.8 157.4±56.2 0.64 ±0.14
28.8±6.5 155.8±52.3 0.64 ±0.17
30.9±6.9 169.7±60.8 0.61 ±0.15
29.8±6.4 165.3±56.5 0.61 ±0.12
F3,54 = 553.14 p= 2.82×10−28 η2= 0.97
p= 2.2×10−5p= 2.1×10−5p= 9.5×10−5
F3,54 = 3.73 p= 0.046 η2= 0.17
F3= 39.82
p= 3.52 ×10−8η2= 0.69
p=
1.01 ×10−7p= 4.49 ×10−5p= 4.99 ×10−8
F3= 41.52
p= 4.73 ×10−14 η2= 0.70
p= 5.44 ×10−7p= 3.25 ×10−9p= 1.46 ×10−5
Rr0(z0, φr) =
cos(φr)−sin(φr) 0
sin(φr) cos(φr) 0
0 0 1
Rhr,1(z0, φh(φr)) =
cos(φh(φr)) −sin(φh(φr)) 0
sin(φh(φr)) cos(φh(φr)) 0
0 0 1
Rhr,2(yh,p, ψh(φr)) =
cos(ψh(φr)) 0 sin(ψh(φr))
0 1 0
−sin(ψh(φr)) 0 cos(ψh(φr))
Rh0(z0, φr+φh(φr)) =
cos(φr+φh(φr)) −sin(φr+φh(φr)) 0
sin(φr+φh(φr)) cos(φr+φh(φr)) 0
0 0 1