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A Minimalist Approach to Biped Walking Robots

David Buckley

@ 4 July 2013

I have been creating Biped Walkers since 1988, starting with the design and operation of the Shadow Biped, and since 1998 have undertaken a development program to build Biped Walkers starting at the other end of the complexity scale. The program so far has resulted in BigFoot; Junior; TecFoot; Sam (Pop), Efi (Mom), Tom, Joe & Kas Amblers; Loki and Freya.

This document has developed from the paper "Experiments with Three Minimal, Autonomous, Biped Walking Robots, David Buckley and Martin Smith, presented at the Climbing and Walking Robots conference CLAWAR 2003". The obfuscation and meaningless references - seemingly neccessary for conference papers - have been removed and further material and the Appendices have been added.

BigFoot   TecFoot   Ambler   Loki  

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Index

Big Foot History
Photos
Mechanics
Dimensions & specification
Control Electronics
Software
Program
Operation

TecFoot History
Static Balance Walking
Dynamic Balance Walking
Mechanics
Dimensions
Control Electronics
Operation in Static Balance
Operation in Dynamic Balance - Active Dynamic Biped Walking using Limit Cycle Control

Ambler History
POP
MOM
Mechanics
Specification
Control Electronics
Operation
Operation on Slopes

Aesir Loki
Mechanics
Operation
Software
Freya
Mechanics
Operation
Further planned robots in the series

References
Appendices Appendix 0 - updates
Appendix 1 - Model Control Servos
Appendix 2 - Software listing for BigFoot
Appendix 3 - SuperTec and hitec Servos
Appendix 4 - Loki Software


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BigFoot
BigFoot by David L Buckley - September 1998

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BigFoot History

Following on from the development of the pneumatically powered Shadow Biped Walker, which incorporates twenty-eight muscles and fifty-six pneumatic valves together with numerous sensors to control the seven degrees of freedom per leg, it was decided to make a much simpler mechanism to eliminate many of the troublesome variables.

Accordingly a frame was designed having parallelogram action legs, each with effectively only one degree of freedom, and feet, again with only one degree of freedom. This frame was equipped with four muscles, four antagonist springs and controlled by a block of eight pneumatic valves. With a little help from a handler this biped walker could easily stroll down the length of the lab.

However, this walker still suffered from a lack of control and so in order to investigate the minimum requirements for true weight shifting biped walking it was decided to make a still simpler model using the same mechanical arrangement but using only two model control type servos (see Appendix-1). So Bigfoot was born.

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BigFoot showing roll and stride positions
stand up roll right stand up right foot forward
Not shown are roll-left and stride with left foot forward.

BigFoot showing three-quarter view and stride-servo
three-quarter view stride-servo

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BigFoot Mechanics

BigFoot, the first of the line is about 22 centimetres high and weighs 450 grams (8.75 inches, 16 ounces) including battery. The frame consists of a body to which are attached two parallelogram action legs and two feet each with one degree of freedom allowing BigFoot to rock from side to side. The top link of each parallelogram is inclined at a slight angle to the ground and the leg members are vertical when the feet are together, the links can only swing forwards and backwards in the sagittal plane. The feet are large so that the length of stride is always less than the length of the foot. The two servos are housed in the body. The stride-servo is placed underneath the roll-servo and the actuator-arm on the output shaft of the stride-servo together with the links to each leg can be seen in the three-quarter view and in the view from underneath. The two legs are each coupled to the stride-servo so that as one moves forwards the other moves backwards. Thus providing that when the legs are in the mid position the centre of mass is arranged to be approximately over the mid point of the feet then for strides of less than the foot length BigFoot is always statically stable in the sagittal plane.

The two feet are attached to the bottom links of the leg parallelograms and can rotate about an axis parallel to the ground and to the fore and aft axis of BigFoot. Again they are coupled to one servo with the linkages arranged so that both feet rotate either clockwise or anticlockwise as viewed from the front of the robot. The disc on the output shaft of the roll-servo together with the wire links to the outer edges of the feet can be seen in the front view. The ankles are placed on the feet so that for any given rotation of a foot the displacement of the lateral edge is greater than that of the medial edge. This ensures that when both feet are rotated together through the same angle about their axes BigFoot will shift its weight from one foot to both then to the other foot.

To turn, BigFoot employs skid steering like a tank, the feet angles are set so that the weight is evenly distributed between the two feet and then one leg is moved forward and the other backwards, then weight is shifted onto the forward foot, the position of the legs reversed, weight shifted to both feet and the process repeated.

The drive links from the servos are arranged so that it is not possible for BigFoot to lean too far in the medial plane and not to make strides of greater than the foot length.

The above arrangement ensures that BigFoot is statically stable at all times apart from a small interval when it is transferring weight from one foot to the other, however this interval is bounded by statically stable positions.

It should be noted that BigFoot (similarly TecFoot and Ambler) are designed for level ground walking and while some variation in levels can be accommodated, a different approach would have been used had they been designed for rough terrain.

BigFoot uses standard size Supertec S03 servos.

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BigFoot dimensions & specification - millimetres (except where noted)

processor - Parallax Basic Stamp BS1
servos    - 2 standard size Supertec S03 servos, rated torque of 3.4Kgcm at 4.8volts [2]
battery   - 4 AA-cells 5 - 6v for servos, and one 9v PP3 for electronics
weight                                  450 grams (16 ounces) including battery.
height overall                         ~220   
height to head excluding circuit board	200
leg length				125
leg horizontal link			 50
height of ankle front link		 18
height of ankle rear link 		 28
width between fore and aft ankle axes 	 30
height of ankle axes 			 10
distance between feet 			 22
width of foot - broadest 		 60
width of foot at heel 			 50
length of foot 				130
rear leg link pivot to back of foot 	 32
maximum stride 				 75
maximum foot lift 			  7
maximum roll	 		      +- 10 degrees
height of Centre of Gravity (CG) 	155
CG from rear of feet (both feet together)55
normal walking stride 			 55
normal walking - steps per minute 	 34
normal walking speed 	   34x55/1000 =   1.87 m/min

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BigFoot Control Electronics

BigFoot is controlled from a Stamp1 system (BS1) available from Parallax Inc. and uses a Microchip Technology Inc. PIC16C56 controller with the user program stored in a 93LC56 EEPROM. Programs are written in a Parallax's high level control-basic language for the BS1, and downloaded from a desktop personal computer (PC).

BigFoot has a tactile sensor on the front of each foot and the control computer had just enough room for routines to step back and turn away when an obstacle was detected.

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BigFoot software

The software listing for the control program used is at Appendix 2.

The program incorporates a command processor which interprets high level robot commands stored in EEPROM at the section ‘user routine‘. Many routines can appear in the listing but due to memory constraints only one can be active. The rest must be commented out. By using this method many different activities can be quickly explored without having to rewrite the program. The program is very simple because of the lack of space in the system used.

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Outline of program

Initialise
----------
initialise robot to stand-up position
set robot-move instruction pointer to start of moves list
Sense
-----
read toe sensors
if a toe sensor is activated then
	set robot-move to turn away from obstacle
	goto case

read next robot-move instruction

case robot-move
	set parameters for robot-move
 end case

set parameters for roll onto foot	: gosub domove
set parameters for move other foot	: gosub domove
set parameters for roll upright		: gosub domove

loop to Sense

sub domove
	do pulse servos
	until move complete
 return

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BigFoot Operation

BigFoot walks by shifting its weight to one foot, making a stride, shifting its weight to the other foot and making the next stride. The method of turning has already been explained above.

BigFoot first walked in November 1998 and immediately proved the concept to be sound and since then many BigFoots have been built and the design copied several times.

Although BigFoot is statically stable as soon as walking at a reasonable speed is attempted it is necessary to take into account the dynamics of the robot.

The main problem is the large mass of the body oscillating from side to side as the weight shifts from one foot to the next. The control system employed, although satisfactory for initial testing of the robot, didn't allow a wide range of speeds to be used for the movement of the legs and feet and the movements are highly rectangular in time rather than sinusoidal. Fortunately the way BigFoot was designed part of this problem was solved by accident. The foot attitude control links bend slightly to absorb energy at the end of a swing, releasing it again as the next cycle starts, in addition to this the whole frame of the robot flexes imperceptibly, storing and releasing energy.

If the servos on BigFoot are operated from a remote control by an operator using visual feedback then it is possible to make BigFoot dance and jig about in a remarkably anamistic way, however the control strategies necessary to do this under computer control are beyond the capabilities of the system employed.


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TecFoot
TecFoot by David L Buckley - July 2000

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TecFoot history

TecFoot was created during the spring and summer of 2000 and builds on the experience gained with BigFoot.
TecFoot started with a design request for a biped five times bigger than BigFoot but using the same two servos. This was impossible since the SO3 servos of BigFoot do not have enough torque. Even using five S03 servos TecFoot is still underpowered for static-balance walking. When operated in quasi dynamic-balance mode TecFoot is much more lively because the servos only pump the oscillations of the system.

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TecFoot in static balance walking configuration


front view

side view

three quarter

normal stride

max. stride


max. lean

max. lean

min. roll (1)

max. roll (2)

(1) min. roll - minimum roll angle to balance on one foot.
(2) max. roll - maximum roll angle when balancing on one foot before overbalancing.


ground clearance during normal walking

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TecFoot in dynamic balance configuration


festooned

showing sprung sole plates

simple switch sensor with visual feedback

general view of foot

TecFoot servos

TecFoot distance sensors

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TecFoot Mechanics

TecFoot is about 63 centimetres high and weighs 1.6 Kilograms (24.5 inches, 4 pound 2 ounces) including batteries and uses five servos, four AA cells for servo power and a 9v battery to power the control electronics.

The frame consists of a body to which are attached two parallelogram action legs and two feet each with one degree of freedom allowing BigFoot to rock from side to side. The top link of each parallelogram is parallel to the ground and when TecFoot is at rest the leg members are vertical, the links can only swing forwards and backwards in the sagittal plane. The feet are large so that the length of stride is always less than the length of the foot. Unlike BigFoot where the top link of the parallelogram is the body itself, the top link is an hip plate which is hinged along a vertical axis, both hip plates are connected to the one servo which alters the angle between the two hip plates (the splay angle) consequently the direction in which the feet point relative to the body and to each other can be changed and so TecFoot does not have to employ skid steer turning.

The forward and backward position of each leg is controlled by a separate servo, unlike BigFoot, so relative to the feet the centre of mass can be shifted forwards and backwards.

The two feet are attached to the bottom links of the leg parallelograms and can rotate about an axis parallel to the ground and to the fore and aft axis of the hip plates. Again, unlike BigFoot they are coupled to separate servos. As there is separate control of each foot the ankles are placed centrally on the feet.

To turn TecFoot shifts its weight over a foot so the other foot is off the ground, alters the splay angle, stands back on two feet, shifts the weight onto the other foot, sets the splay angle to zero and stands back on two feet or takes a step.

The above arrangement ensures TecFoot is generally statically stable at all times but because the centre of mass can be shifted forwards and backwards care must be taken during a turn to avoid overbalancing forwards or backwards.

TecFoot uses standard size Supertec S03 servos.

The S03 servos used to tilt the feet have only just enough torque to hold TecFoot balancing on one foot when run off a battery of 4-AA NiCad cells. However by operating TecFoot so that the balance point is never reached and the action is to always fall towards the other foot safe operation is ensured even when the battery is running down. Both TECFOOT.BS2 and DYNAMIC.BS2 use this method.

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TecFoot dimensions & specification - millimetres (except where noted)

processor - Parallax Basic Stamp BS2 + 2 Milford Servo co-processors
servos    - 5 standard size Supertec S03 servos, rated torque of 3.4Kgcm at 4.8volts [2]
battery   - 4 AA-cells 5 - 6v for servos, and one 9v PP3 for electronics
weight      (including battery)                         1.6 Kg (16 ounces) 
height overall                                         ~630 (24.5 inch)  
height to head excluding circuit board			610
leg length						500
leg horizontal link					 55
height of hip 						560
distance between hip vertical axes 			120
distance of front leg-link to hip vertical axis 	 20
width between fore and aft ankle axes 			160
height of ankle axes 					 20
distance between feet (feet parallel) 			 60
width of foot						120
length of foot 						220
rear leg link pivot to back of foot			 35
maximum stride 						130
minimum roll (static balance) 		              +- 10 degrees
maximum roll (static balance)                         +- 20 degrees
minimum foot lift (static balance)              	 15
maximum foot lift  					 35
CG from rear of feet (normal walking) 			110
height of Centre of Gravity (CG) 			400
normal walking stride (static balance) 			100
normal walking (static balance)- steps/minute 	 	 24
normal walking speed (static balance)    24x100/1000= 	  2.4 m/min
normal walking (dynamic balance) - steps/minute 	144
maximum roll (dynamic balance)                        +- 10 degrees
normal walking stride (dynamic balance)
			stride is dynamically adjusted 45 - 70
			average				 50
normal walking speed (dynamic balance)
					     144x50/1000= 7.2 m/min

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TecFoot Control Electronics

TecFoot is controlled from a Stamp2 system (BS2) available from Parallax Inc. which uses a Microchip Technology Inc. PIC16C57 controller with the user program stored in 2k bytes of EEPROM. Programs are written in Parallax's high level control-basic language for the BS2 and downloaded from a desktop personal computer (PC).

To relieve the processor of sending pulses to the five servos every twenty milliseconds (see Appendix-1) two co-processors are used each capable of controlling three servos. The co-processors are sent instructions over serial links at 9600 baud.

Each foot has four underfoot sprung tactile sensors, originally simple cheap momentary action push to make single pole switches now upgraded to microswitches.

Two Parallax Ping ultrasonic distance sensors are fitted and read by the BS2 to detect obstacles A PicAxe18m2 auxiliary processor is used to read two Sharp Infra-red distance sensors aimed at the floor in front odf the toes to detect cliffs, drop offs, steps down and up and other close obstacles. The PicAxe18m2 also drives some indicator LEDs.

The BS2 also uses a two line by 16 character LCD to show the operational state.

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TecFoot Operation

TecFoot walks by shifting its weight to one foot, making a stride, shifting its weight to the other foot and making the next stride. The method of turning has already been explained above.The parameters for fore and aft posture, stride length, speed, roll angle and splay angles are fixed at compile time and it is up to the operator to adjust them to suitable values. In Dynamic operation the roll angle is automatically adjusted in software during operation.

TecFoot has separate control over the horizontal angle each foot and to avoid having to calculate the exact angle of each foot when shifting its weight from one foot to the other the mechanical control links from the foot servos to the feet are arranged to act more or less as tendons and have only a limited ability to push. The link is designed to bend when under compression.

As the underfoot tactile sensors are sprung TecFoot is able to use them to determine whether its weight is evenly distributed over each foot and adjust its posture as appropriate by swaying forwards or backwards until all four switches are closed. If the toes or heels are only lightly loaded then there will not be enough pressure to overcome the spring in the toe or heel switches.

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TecFoot Operation - static balance mode

Operation of TecFoot in Static Balance mode is very similar to that for BigFoot described above except that TecFoot having smaller feet relative to its height is less tolerant to imperfections in the surface on which it walks. While it is stable on smooth wood floors and thin or stiff pile carpets it is less so on shag pile and tamped concrete. All parameters for stride length, speed, roll angle are fixed at compile time and it is up to the operator to adjust them to suitable values. When this is done the walking stable and fluid.

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TecFoot Operation - dynamic balance mode
- Active Dynamic Biped Walking using Limit Cycle Control.

With the incorporation of the underfoot tactile sensors it was possible to make TecFoot work in a quasi dynamic balance mode, that is, TecFoot rocks dynamically from one foot to the other whilst walking, testing the sensors to see when to change states and altering the rock parameters to stay within the limit cycle. The amount of energy injected into the system on each step is altered so that stable oscillation is maintained.

As soon as this was tried it was apparent that TecFoot in this mode was much more stable coping with shag pile carpets and walking on and off rugs with ease.

In static balance mode the limits of stability when balancing on one foot are +10 to +20 degrees roll off-centre, that is TecFoot is required to balance within +-5 degrees after rolling a minimum of 20 degrees from standing on the other foot. However in dynamic mode the basin of stability is the whole of the 20degrees roll from one foot to the other. Not only was TecFoot much more stable but walking was now three times faster (7.2m/min / 2.4m/min).


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Ambler
Ambler by David L Buckley - September 2002

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Ambler History

One of the copies of BigFoot, mentioned above, was the Toddler robot from Parallax Inc. It walks in a very stiff way but it is a better walker than BigFoot. This prompted a re-evaluation of BigFoot which resulted in the Ambler design. Ambler was designed in the summer of 2002.

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Pop


at rest


normal walking roll 8 deg.


maximum roll 38 deg.

at rest


normal walking stride 55mm


rear view

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Mom


at rest

normal walking roll

maximum roll 28 deg.

at rest


normal walking stride 32mm


maximum stride 65mm

three-quarter view

servo placement

stride-servo and links

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Ambler Mechanics

Ambler, the third of the line using the parallelogram leg design is about 24 centimetres high and weighs 560 grams (9.5 inches 20 ounces) including batteries and uses two servos, four AA cells for servo power and a 9v battery to power the control electronics.

Five Amblers have now been built all with slightly different geometries, the first has the batteries in the feet and is named Pop (aka. Sam) because it looked to have a fat belly, consequently the second had to be Mom (aka. Efi). The third is Tom, taller with three servos; the fourth is Joe, nearly the same as Sam; while the fifth is Kas, the same height as Sam but with three servos. Tom, Joe and Kas will be described later.

As with BigFoot the frame consists of a body housing two servos, attached to the body are two parallelogram action legs, at the lower end of each leg is an ankle which is the lower horizontal link and a foot. Each foot has one degree of freedom allowing Ambler to rock from side to side. The positioning of the two servos is the same as in BigFoot except that the output shaft of the roll-servo is at the back of the robot. The top link of each leg parallelogram is parallel to the ground and the leg members are vertical when both feet are together, the links can only swing forwards and backwards in the vertical plane. The feet are large so that the length of stride is always less than the length of the foot. The two legs are each coupled to the same servo so that as one moves forwards the other moves backwards. Thus providing that when the legs are in the mid position the centre of mass is arranged to be approximately over the mid point of the feet then in the fore and aft plane Ambler is always statically stable.

The two feet are attached to the bottom links of the leg parallelograms and can rotate about an axis parallel to the ground and to the fore and aft axis of Ambler. Again they are coupled to one servo with the linkage arranged so that both feet rotate either clockwise or anticlockwise as viewed from the front of the robot. The ankles are placed on the feet so that for any given rotation of a foot the displacement of the lateral edge is greater than that of the medial edge. This ensures that when both feet are rotated together through the same angle about their axes BigFoot will shift its weight from one foot to both then to the other foot.

To turn Ambler employs skid steering like a tank, the feet angles are set so that the weight is evenly distributed between the two feet and then one leg is moved forward and the other backwards, then weight is shifted onto the forward foot, the position of the legs reversed, weight shifted to both feet and the process repeated.

The above arrangement ensures Ambler is statically stable at all times apart from a small interval when it is transferring weight from one foot to the other, however this interval is bounded by statically stable positions.

The design of Ambler is improved over that for BigFoot in several respects.

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Ambler dimensions & specification - millimetres (except where noted)

(values for Mom only where different) Pop Mom ------------------------------------- --- --- processor - Parallax Basic Stamp BS2 servos - 2 standard size Supertec S03 servos, rated torque of 3.4Kgcm at 4.8volts [2] battery - 4 AA-cells 5 - 6v for servos, and one 9v PP3 for electronics height overall ~240 (9.5 inches) mass (gram) (including battery) 547 with underfoot switches and PicAxe28 processors mass (gram) 509 with underfoot switches height to head excluding circuit board 218 Leg length 125 leg horizontal link 40 30 height of ankle joints of leg links 20 width between fore and aft ankle axes 50 height of ankle axes 15 distance between feet 10 width of foot - broadest 65 length of foot 110 rear leg link pivot to back of foot 35 maximum stride 65 maximum foot lift 23 18 maximum roll - degrees +- 38 28 Height of Centre of Gravity (CG) 108 154 CG from rear of feet (both feet together 55 Foot lift for normal walking 5 Normal walking stride 55 32 Normal walking - steps per minute 72 104 Maximum steps per minute 80 104 (POP falls over at 80spm & 55mm stride) Normal walking speed - POP - 72x55/1000 = 3.96 m/min Normal walking speed - MOM - 104x32/1000 =~3.33 m/min AA Alkaline cell capacity 2500mAh Average current while walking 200mA => endurance on one set of AA Alkaline cells >12 hours or endurance on one set of AA NiCad 1.2Ah cells >6 hours

Amblers use standard size Supertec S03 servos.

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Ambler Control Electronics

As with TecFoot the Amblers are controlled from a Stamp2 system available from Parallax Inc. which uses a Microchip Technology Inc. PIC16C57 controller with program stored in 2k bytes of EEPROM. Programs are written in a high level language and downloaded from a desktop personal computer (PC).

The Amblers have a tactile sensor on the front of each foot which allow the control computer to initiate a step back and turn away when an obstacle is detected.

Besides the toe sensors the Amblers are equipped with a plugin breadboard area, expansion bus, I2C port, status LEDs, piezo transducer for easy remote monitoring of program operation and a user potentiometer useful for setting the speed or other operational parameters.

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Ambler Operation

Where Pop has two AA cells in each foot Mom has the cells at the top of the body, putting the cells in the feet of Pop helped with the foot link tension but meant running the power lines up the legs and Mom was built as an experiment to see how the improved geometry of Pop would work with the mass of the battery at the top of the robot rather than the bottom.

Mom obviously has more inertial mass in the body because of the placement of the AA cells but providing the parameters for the step length and roll are suitable adjusted then Mom is able to walk nearly as fast as Pop, Pop is able to take long strides while Mom needs to take shorter quicker steps relying more on the flexing of the body to store and release energy. One onsequence of the energy transfer in Mom, and the fact that at each step she doesn’t have to pick up the two AA cells that POP has in his feet, is that the batteries last slightly longer than those in Pop.

At slow speed the Amblers walk using predefined gaits, at fast speed, as mentioned above, the frame and tilt rods, because they are flexible, store and release energy throughout the walking cycle, consequently the Amblers are then Active Dynamic Biped Walkers with Limit Cycle Control.

Both Mom & Pop Amblers are now very good stable walkers with POP being able to cope with variations in level of up to six millimetres and slopes of up to about ten degrees fore and aft but only 4 degrees sideways.

It will be seen from comparing the ‘dimensions & specification’ tables for BigFoot and Ambler that the Amblers are able to walk about twice as fast as BigFoot.

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Operation on slopes - photos show maximum static positions

Pop

max slope 15 deg.

max slope 15 deg.

max slope 15 deg.

max slope 4 deg.

Mom

max slope 6 deg.

max slope 9 deg.

max slope 9 deg.

max slope 4 deg.

For both Pop and Mom, for weight-transfer to be able to take place from the downhill foot onto the uphill foot, the limiting slope is four degrees.

At slopes of over fifteen degrees Pop slides down but since it is still limited by the 4 maximum transfer slope the fitting of soles with higher friction was not considered useful.

At actual practical walking speeds the figures are slightly different due to the inertial forces experienced.

Dynamic operation up/down slope - at a step length of 35 millimetres

Pop 12 deg. up 12 deg. down
Mom  8 deg. up 10 deg. down

Dynamic operation along slope - at a step length of 35 millimetres

Pop 4 deg.
    6 deg. with dynamic swing from downhill foot to uphill foot
   12 deg. just dragging the downhill foot
Mom 4 deg.
    5 deg. with dynamic swing from downhill foot to uphill foot
    6 deg. just dragging the downhill foot

If a quick push off is made with the downhill foot then the impetus can enable the robot to roll past the equilibrium point and come to rest on the opposite foot. This only makes two degrees difference for POP and one degree for MOM.


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Aesir series

(The Aesir are a group of Norse gods.)[1]
This is a series of Biped Walkers each using four servos, two servos per leg. These will explore the abilities of the different geometries obtainable using just two servos per leg and hence just two degrees of freedom per leg.
The robots are much squatter than those previously mentioned but, because of the extra two servos, have far greater agility.

Loki

The goals of the Aesir project are for them to be able to play with a ball and collect bricks.

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Aesir Control Electronics
The Aesir series uses the same electronics as the Amblers

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Loki
Loki by David L Buckley - December 2002
Loki is the first of the series and was designed in late 2002.

Loki at rest

Loki stepping high

Loki kicking high

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Loki Mechanics
Loki is 16 centimetres high by 17 centimetres wide by 13 centimetres front to back and weighs 620 grams including batteries (6.25" x 6.75" x 5.25", 22.5oz)
Each leg at the hip is pivoted around a vertical axis while the foot is pivoted about an horizontal fore and aft axis.
The distance between the ankle pivots is 130mm and so to lift one foot off the ground the torque required is:

620/1000*13/2=4.03Kgcm
A standard size Supertec S03 servo has a rated torque of 3.4Kgcm at 4.8volts [2].
A standard size Supertec S06/2BB servo has a rated torque of 7.2Kgcm at 4.8volts [3].
Loki uses standard size Supertec S03 servos at the hips and the higher torque Supertec S06 servos at the ankles.

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Loki Operation
One consequence of the design is that because Loki does not use weight shifting the ankle servos need to be more powerful than the servos used in BigFoot and Ambler.
Loki originally was able to self-right and stand-up unaided from all positions except directly upside down. The addition of roll bars on the head so Loki would automatically roll from being upside down to lying on its front or back would have prevented that exception. However to enable Loki to step and turn more easily the toes were shortened and now Loki is not able to self-right.

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Loki Software
Appendix 4 - Loki Software

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Freya
Freya by David L Buckley - December 2002
(Freya was not actually one of the Aesir but was married to Odin who was.)
...Freya is one of the foremost goddesses of the Vanir...she married the mysterious god Od (probably another form of Odin)...http://www.pantheon.org/articles/f/freya.html
Freya is the second of the series and was designed in late 2002.

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Freya Mechanics
Freya is 17.5 centimetres high by 20 centimetres wide by 19 centimetres front to back and weighs 620 grams including batteries (6.25" x 6.75" x 5.25", 22.5oz).
Each leg is pivoted at the hip around a fore and aft horizontal axis while the foot is pivoted about a vertical axis relative to the foot when on the ground.
The distance between the hip pivots is 135mm and so to lift one foot off the ground the torque required is:

620/1000*13.5/2 = 4.185Kgcm
A standard size Supertec S03 servo has a rated torque of 3.4Kgcm at 4.8volts [2]
A standard size Supertec S06/2BB servo has a rated torque of 7.2Kgcm at 4.8volts [3]
Freya uses standard size Supertec S03 servos at the ankles and higher torque Supertec S06 servos at the hips (the reverse of Loki).

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Freya Operation
One consequence of the design is that because Freya does not use weight shifting the hip servos need to be more powerful than the servos used in BigFoot and Ambler
Freya is able to self-right and stand-up unaided from all positions.

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Further planned robots in the series
Thor and Ulur will keep their bodies horizontal and lift their feet.
Frey (the brother of Freya) will be a more robust development of Freya and use a different foot design.
Odin will use weight shifting.

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References:

[1] Everyman's Dictionary of Non-Classical Mythology - Egerton Sykes, Dent 1961

[2] Appendix 3, SuperTec S03 servo specification from J Perkins

[3] Appendix 3, SuperTec S06 servo specification from J Perkins


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Appendix 0 - updates

Appendix 1 - Model Control Servos

Appendix 2 - Software listing for BigFoot

Appendix 3 - SuperTec and hitec Servos